





• 


LIBRARY OF CONGRESS. 


1 


®^pTIlE4 ®ijp5ri# If a 

Shelf ...'..S3, "i 
1552, 

UNITED STATES OF .iMEEICA. 







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t ^ g 

1/5 5 

CO jH 





FORMULAS 



IN 



GEARING 



"^"X. 



WITH PRACTICAL SUGGESTIONS 



f3 



s 




PROVIDENCE, R. I. 

BROWN & SHARPE MANUFACTURING COMPANY. 



1893. 






Entered according to Act of Congress, in the year 1892 by 

BROWN & SHARPE MFG. CO., 

In the Office of the Librarian of Congress at Washington. 

Registered at Stationers' Hall, London, Eng. 

AH rights reserved. 



A 



&^ 



'vK 



Olo -3'/5'i'?' 



PREFACE. 

It is the aim, in the following pages, to condense as much 
as possible the solution of all problems in gearing which in the 
ordinary practice may be met with, to the exclusion of prob- 
lems dealing with transmission of power and strength of 
gearing. The simplest and briefest being the symbolical 
expression, it has, whenever available, been resorted to. The 
mathematics employed are of a simple kind, and will present 
no difficulty to anyone familiar with ordinary Algebra and 
the elements of Trigonometry. 



CONTENTS. 

FORMULAS IN GEARING. 



CHAPTER I. 

Page 

Systems of Gearing . . . . . i 

CHAPTER II. 

Spur Gearing — Formulas — Table of Tooth Parts — Comparative Sizes 

of Gear Teeth 4 

CHAPTER III. 

Bevel Gears, Axes at Right Angles — Formulas — Bevel Gears, Axes at 
any Angle— Formulas — Undercut in Bevel Gears — Diameter Incre- 
ment — Tables for Angles of Edge and Angles of Face — Tables of 
Natural Lines 1 1 

CHAPTER IV. 
Worm and Worm Wheel, Formulas — Undercut in Worm Wheels 32 

CHAPTER V. 

Spiral or Screw Gearing — Axes Parallel — Axes at Right Angles — Axes 

at any Angle — General Formulas 36 

CHAPTER VI. 
Internal Gearing — Internal Spur Gearing — Internal Bevel Gears 41 

CHAPTER VII. 
Gear Patterns 47 

CHAPTER VIII. 
Dimensions and Form for Bevel Gear Cutters 50 

CHAPTER IX. 
Directions for cutting Bevel Gears with Rotary Cutter 53 

CHAPTER X. 
The Indexing of any Whole or Fractional Number 56 

CHAPTER XI. 

The Gearing of Lathes for Screw Cutting — Simple Gearing — Compound 

Gearing — Cutting a Multiple Screw 60 



FORMULAS IN GEARING. 



CHAF'TE^R I. 

SYSTEMS OF GEARING. 

(Figs. I, 2.) 

There are in common use two systems of gearing, viz.: the 
involute and the epicycloidal. 

In the involute system the-outlines of the working parts of a 
tooth are single curves, which may be traced by a point in a 
flexible, inextensible cord being unwound from a circular disk 
the circumference of which is called the base circle^ the disk 
being concentric with the pitch circle of the gear. 




In Fif;;. i the two base circles are represented as tangent to 
the line P P. This line (P P) is variously called *' the line of 
pressure," " the line of contact," or '' the line of action." 



2 BROWN & SHARPE MFG. CO. 

In our practice this is drawn so as to make with a normal 
to the center line (O O') 14%°, or with the center line 75^°. 

The rack of this system has teeth with straight sides, the two 
sides of a tooth making, together, an angle of 29° (twice 

14)^°). 

This applies to gears having 30 teeth or more. For gears 
having less than 30 teeih special rules are followed, which are 
explained in our " Practical Treatise on Gearing." 




Fig. 2. 



In eptcycloidal, or double-curve teeth, the formation of the 
curve changes at the pitch circle. The outline of the faces of 
epicycloidal teeth may be traced by a point in a circle rolling 
on the outside of pitch circle of a gear, and the flanks by a point 
in a circle rolling on the inside of the pitch circle. The faces 
of one gear must be traced by the same circle that traces the 
flanks of the engaging gear. 

In our practice the diameter of the rolling or describing 
circle is equal to the radius of a 15-tooth gear of the pitch 
required ; this is the base of the system. The same describing 
circle being used for all gears of the same pitch. 



PROVIDENCE, R. I. 3 

The teeth of the rack of this system have double curves, 
which may be traced by the base circle rolling alternately on 
each side of the pitch line. 

An advantage of the involute over the epicycloidal tooth is, 
that in action gears having involute teeth may be separated a 
little from their normal positions without interfering with the 
angular velocity, which is not possible in any other kind of 
tooth. 

The obliquity of action is sometimes urged as an objection 
to involute teeth, but a full consideration of the subject will 
show that the importance of this has been greatly over-esti- 
mated. 

The tooth dimensions for both the involute and epicycloidal 
gears may be calculated from the formulas in Chapter II. 



BROWN & SHARPE MFG. CO. 



CHAPTER II. 



SPUR GEARING. 

(Figs. 3, 4.) 

Two spur gears in action are comparable to two correspond- 
ing plain rollers whose surfaces are in contact, these surfaces 
representing the pitch circles of the gears. 

Pitch of Gears. 

For convenience of expression the pitch of gears may be 
stated as follows : 

Circular pitch is the distance from the center of one tooth to 
the center of the next tooth, measured on the pitch line. 

Diametral pitch is the number of teeth in a gear per inch of 
pitch diameter. That is, a gear that has, say, six teeth for each 
inch in pitch diameter is six diametral pitch, or, as the expres- 
sion is universally abbreviated, it is " six pitch." This is by 
far the most convenient way of expressing the relation of 
diameter to number of teeth. 

Chordal pitch is a term but little employed. It is the dis- 
tance from center to center of two adjacent teeth measured in 
a straight line. 



-r 




Fig. S. 



(^t—^ 



PROVIDENCE, R. I. 



FORMULAS. 



N = number of teeth. 
s = addendum. 

/ = thickness of tooth on pitch line. 
/= clearance at bottom of tooth. 
D" = working depth of tooth. 
D" + / = whole depth of tooth. 
d = pitch diameter. 
d' = outside diameter. 
P' = circular pitch. 
P^ = chord pitch. 
P = diametral pitch. 
C = center distance. 



._N + 


2 


d' 




, 7t 
P' 




„ _7t 
P 




I 


P' 



.= _L=^ = .3r83P' 

d d' 

s = 



N N + 2 

2 2 P 



lO 



lO 
s + 
Ti" — 2S 

yc — d sin 

N 

360^ 



6 P"^ 

V =. dn where sin d = — 



d-- 

d' := d -h 2 S 
7t 



BROWN & SHARPE MFG. CO. 



GEAR WHEELS. 

TABLE OP TOOTH PARTS CIRCULAR PITCH IN FIRST COLUMN. 



p 


Threads or 

Teeth per inch 

Linear. 


11 
ft 


Thickness of 

Tooth on 
Pitch Line. 


a 




^ Depth of Space 
+ below 
'^ Pitch Line. 


Whole Depth 
of Toolh. 


Width of 

Thread-Tool 

at End. 


o 


J " 


p 


t 


8 


D" 


D"+/ 
1.3732 


P'x.31T'x.335 


2 


i 


1.5708 


1.0000 


.6366 


1.2732 


.7366 


.6200 


.6700 


1| 


A 


1.6755 


.9375 


.5968 


1.1937 


.6906 


1.2874 


.5813 


.6281 


If 


t 


1.7952 


.8750 


.5570 


1.1141 


.6445 


1.2016 


.5425 


.5863 


Ifi 


A 


1.9333 


.8125 


.5173 


1.0345 


.5985 


1.1158 


.5038 


.5444 


li 


1 


2.0944 


.7500 


.4775 


.9549 


.5525 


1.0299 


.4650 


.5025 


h\ 


11 


2.1855 


.7187 


.4576 


.9151 


.5294 


.9870 


.4456 


.4816 


If 


A 


2.2848 


.6875 


.4377 


.8754 


.5064 


.9441 


.4262 


.4606 


iiV 


a 


2.3936 


.6562 


.4178 


.8356 


.4834 


.9012 


.4069 


.4397 


li 


i 


2.5133 


.6250 


.3979 


.7958 


.4604 


.8583 


.3875 


.4188 


lA 


H 


2.6456 


.5937 


.3780 


.7560 


.4374 


.8156 


.3681 


.3978 


H 


1 


2.7925 


.5625 


.3581 


.7162 


.4143 


.7724 


.3488 


3769 


ItV 


H 


2.9568 


.5312 


.3382 


.6764 


.3913 


.7295 


.3294 


.3559 


1 


1 


3.1416 


.5000 


.3183 


.6366 


.3683 


.6866 


.3100 


.3350 


a 


lA 


3.3510 


.4687 


.2984 


.5968 


.3453 


.6437 


.2906 


.3141 


i 


H 


3.5904 


.4375 


.2785 


.5570 


.3223 


.6007 


.2713 


.2931 


H 


lA 


3.8666 


.4062 


.2586 


.5173 


.2993 


.5579 


.2519 


.2722 


f 


1* 


4.1888 


.3750 


.2387 


.4775 


.2762 


.5150 


.2325 


.2513 


ii 


lA 


4.5696 


.3437 


.2189 


.4377 


.2532 


.4720 


.2131 


.2303 


1 


li 


4.7124 


.3333 


.2122 


.4244 


.2455 


.4577 


.2066 


.2233 



PROVIDENCE, R. I. 

TABLE OF TOOTH FABT^.— Continued. 

CIECULAR PITCH IN FIRST COLUMN. 



11 

P- 


Threads or 

Teeth^ per inch 

Linear. 


si 


Thickness of 

Tooth on 

Pitch Line. 




k 

iiS 

1^ 


Depth of Space 

below 

Pitch Line. 


k 


Width of 

Thread-Tool 

at End. 


Width of 
Thread at Top. 


1 " 


p 


t 


s 


D" 


^-h/ 


D'M-/. 


Px.31 


Fx.335 


1| 


5.0265 


.3125 


.1989 


.3979 


.2301 


.4291 


.1938 


.2094 


A 


1^ 


5.5851 


.2812 


.1790 


.3581 


.2071 


.3862 


.1744 


.1884 


i 


2 


6.2832 


.2500 


.1592 


.3183 


.1842 


.3433 


.1550 


.1675 


r\ 


2f 


7.1808 


.2187 


.1393 


.2785 


.1611 


.3003 


.1356 


.1466 


f 


2i 


7.8540 


.2000 


.1273 


.2546 


.1473 


.2746 


.1240 


.1340 


f 


2f 


8.377G 


.1875 


.1194 


.2387 


.1381 


.2575 


.1163 


.1256 


* 


3 


9.4248 


.1666 


.1061 


.2122 


.1228 


.2289 


.1033 


.1117 


i\ 


^ 


10.0531 


.1562 


.0995 


.1989 


.1151 


.2146 


.0969 


.1047 


\ 


3^ 


10.9956 


.1429 


.0909 


.1819 


.1052 


.1962 


.0886 


.0957 


i 


4 


12.5664 


.1250 


.0796 


.1591 


.0921 


.1716 


.0775 


.0838 


1 


^i 


14.1372 


.1111 


.0707 


.1415 


.0818 


.1526 


.0689 


.0744 


\ 


5 


15.7080 


1000 


.0637 


.1273 


.0737 


.1373 


.0620 


.0670 


IC 


^4 


16.7552 


.0937 


.0597 


.1194 


.0690 


.1287 


.0581 


.0628 


i 


6 


18.8496 


.0833 


.0531 


.1061 


.0614 


.1144 


.0517 


.0558 


\ 


7 


21.9911 


.0714 


.0455 


.0910 


.0526 


.0981 


.0443 


.0479 


i 


8 


25.1327 


.0625 


.0398 


.0796 


.0460 


.0858 


.0388 


.0419 


I 


9 


28.2743 


.0555 


.0354 


.0707 


.0409 


.0763 


.0344 


.0372 


,\ 


10 


31.4159 


.0500 


.0318 


.0637 


.0368 


.0687 


.0310 


.0335 


t\ 


16 


50.2655 


.0312 


.0199 


.0398 


.0230 


.0429 


.0194 


.0209 



BROWN & SHARPE MFG. CO, 

GEAR WHEELS. 

TABLE OF TOOTH PAETS DIAMETRAL PITCH IN FIEST COLUMN. 



1 . 

II 

1 


ll 


Thickness 
of Tooth on 
Pitch Line. 


S 

ro a 

< 


% 

o ^ 

1^ 


Depth of Space 

below 

Pitch Line. 




P 


P' 


t 


s 


D' 


s+f. 


D"+f. 


i 


6.2832 


3.1416 


2.0000 


4.0000 


2.3142 


4.3142 


2. 

A 


4.1888 


2.0944 


1.3333 


2.6666 


1.5428 


2.8761 


1 


3.1416 


1.5708 


1.0000 


2.0000 


1.1571 


2.1571 


li 


2.5133 


1.2566 


.8000 


1.6000 


.9257 


1.7257 


li 


2.0944 


1.0472 


.6666 


1.3333 


.7714 


1.4381 


If 


1.7952 


.8976 


.5714 


1.1429 


.6612 


1.2326 


2 


1.5708 


.7854 


.5000 


1.0000 


.5785 


1.0785 


2i 


1.3963 


.6981 


A4A4. 


.8888 


.5143 


.9587 


2i 


1.2566 


.6283 


.4000 


.8000 


.4628 


.8628 


2f 


1.1424 


.5712 


.3636 


.7273 


.4208 


.7844 


3 


1.0472 


.5236 


.3333 


.6666 


.3857 


.7190 


3i 


.8976 


.4488 


.2857 


.5714 


.3306 


.6163 


4 


.7854 


.3927 


.2500 


.5000 


.2893 


.5393 


5 


.6283 


.3142 


.2000 


.4000 


.2314 


.4314 


G 


.5236 


.2618 


.1666 


.3333 


.1928 


.3595 


7 


.4488 


.2244 


.1429 


.2857 


.1653 


.3081 


8 


.3927 


.1963 


.1250 


.2500 


.1446 


.2696 


9 


.3191 


.1745 


.1111 


.2222 


.1286 


.2397 


10 


.3142 


.1571 


.1000 


.2000 


.1157 


.2157 


11 


.2856 


.1428 


.0909 


.1818 


.1052 


.1961 


12 


.2618 


.1309 


0833 


.1666 


.0964 


.1798 


13 


.2417 


.1208 


.0769 


.1538 


.0890 


.1659 


14 


.2244 


.1122 


.0714 


.1429 


.0826 


.1541 



PROVIDENCE, R. I. 

TABLE OF TOOTH TABTS— Continued. 

DIAMETRAL PITCH IN FIRST COLUMN. 



is 

Q 


H 


Thickness 
of Tooth on 
Pitch Line. 


1- 


.rl 

to 


Depth of Space 

below 

Pitch Line. 


Is 


P. 


P'. 


t. 


s. 


D'. 


s+f- 
.0771 


D"-^/. 


15 


.2094 


.1047 


.0666 


.1333 


.1438 


16 


.1963 


.0982 


.0625 


.1250 


.0723 


.1348 


17 


.1848 


.0924 


.05-^8 


.1176 


.0681 


.1269 


18 


.1745 


.0873 


.0555 


.1111 


.0643 


.1198- 


19 


.1653 


.0827 


.0526 


.1053 


.0609 


.1135 


20 


.1571 


.0785 


.0500 


.1000 


.0579 


.1079 


22 


.1428 


.0714 


.0455 


.0909 


.0526 


.0980 


24 


.1309 


.0654 


.0417 


.0833 


.0482 


.0898 


26 


.1208 


.0604 


.0385 


.0769 


.0445 


.0829 


28 


.1122 


.0561 


.0357 


.0714 


.0413 


.0770 


30 


.1047 


.0524 


.0333 


.0666 


.0386 


.0719 


32 


.0982 


.0491 


.0312 


.0625 


.0362 


.0674 


34 


.0924 


.0462 


. .0294 


.0588 


.0340 


.0634 


36 


.0873 


.0436 


.0278 


.0555 


.0321 


.0599 


38 


.0827 


.0413 


.0263 


.0526 


.0304 


.0568 


40 


.0785 


.0393 


.0250 


.OoOO 


.0289 


.0539 


42 


.0748 


.0374 


.0238 


.0476 


.0275 


.0514 


44 


.0714 


.0357 


.0227 


.0455 


.0263 


.0490 


46 


.0683 


.0341 


.0217 


.0435 


.0252 


.0469 


48 


.0654 


.0327 


.0208 


.0417 


.0241 


.0449 


50 


.0628 


.0314 


.0200 


.0400 


.0231 


.0431 


56 


.0561 


.0280 


.0178 


.0357 


.0207 


.0385 


60 


.0524 


.0262 


.0166 


.0333 


.0193 


.0360 



lO 



BROWN & SHARPE MFG. CO. 



Comparative Sizes of Gear Teeth. 
Involute. 




8 p 



Fig. 4, 



9 P 



PROVIDENCE, R. 1. 



II 



CHAPTER III. 

BEVEL GEARS.— AXES AT RIGHT ANGLES. 

(Fig. 5.) 




12 BROWN & SHARPE MFG. CO. 



FORMULAS. 

^« ^ t Number of teeth -j ^?^T' 
Nj, = j ( pinion 

P = diametral pitch. 

P' = circular pitch. 

center angle = angle of edge j gear, 
or pitch angle ( pinion. 

/3 = angle of top. 

/3' = angle of bottom. 

^« "^ I angle of face \ ^?^^' 
gb= ) ^ I pmion. 



0!b = \ 



," >■ cutting: angle < ^. . 



/if, = ) & & j pinion. 

A = apex distance from pitch circle. 
A' = apex distance from large bottom of tooth. 
^= pitch diameter. 
d' = outside diameter. 
s = addendum. 

/ = thickness of tooth at pitch line. 
/= clearance at bottom of tooth. 
D" = working depth of tooth. 
D" + /= whole depth of tooth. 
2 a = diameter increment. 
d = distance from top of tooth to plane of pitch circle. 
F = v^ridth of face. 



PROVIDENCE, R. I. 13 



tan a^ =—~— ; tan a^, = / ; 

^ 2 sin a 4. /3 s 

tan p = ; or tan = — • 

^ N ' ^ A 

' N N ' ^ A ' 

/i = a — /J' (See Note, page §2.) 
2 



*-v©'Msy 



A= " 



2 P sin a 



A'= _^ A' - N 



COS fi' 2 P sin « cos /!^' 

A = i^^ cos Q 

Sin (a + /^) ^ 

P= N 



2 A sin a 



^ = — or == ^ ■= d ■\- 2 a 

P ;r 

2^=2^ cos a {See page 20.) 

, , \ a for 2:ear = /^ for pinion 

^ = «tana j ^ f^r pinion = Mor gear 

P=_5_ P'=-^ 

P' P 

J = 2 =^ = .3183 P' j = Atan^ 
^+/= .3685 P' s +/=AtSin/3' 

F = i + - or = 2 P' to 3 P' 
P 7 

Note.— Formulas containin^^ notations without the designating letters a and d 
apply equally to either gear or pinion. If wanted for one or the other, the respective 
letters are simply attached. 



14 



BROWN & SHARPE MFG. CO. 



BEVEL GEARS WITH AXES AT ANY ANGLE. 



(Figs. 6, 7.) 



' I*inion 




Fig. 6. 



PROVIDENCE, R. I. 15 



FORMULAS. 

C = angle formed by axes of gears. 

J^" "^ I number of teeth \ ^?^F' 
rif, = ) { pinion. 

P = diametral pitch. 

P' = circular pitch. 

^l Z [ angle of edge = pitch angle | ^f^^kin. 
/3 = angle of top. 
/3' = angle of bottom. 

^" ^ I angle of face \ ^?^^' 
^b= ) ^ l pmion. 

j; "^ I cutting angle \ ^?^^ 
/lb = ) & & ^ pinion. 

A = apex distance from pitch circle. 
A' = apex distance from large bottom of tooth. 

d = pitch diameter. 
d' = outside diameter. 
2a = diameter increment. 

l^ = distance from top of tooth to plane of pitch circle. 



Note. — The formulas for tooth parts as given on page 5 apply equally to these 
cases. 



l6 BROWN & SHARPE MFG. CO. 



sin C ^ Nft . ^ ^ 

tan «'„= — ; or cot a^ = — — i^-— + cot C 

^'' + cosC N"^^"^ 

Na 

tan rt'b = — — — ; or cot a^ = — — ;^^—- 4- cot C 

^1 + cos C N. s.n C 

Note.— These formulas are correct only for values of C less than 90°. If C 
greater than 90°, consult the following page. 



., 2 Sin a 4, a S 

tan p = ; or tan a = — ; 

' N A ' 

tan fi' = ^^" ""(^ + T%) = 2.3iyin ^ . ^an yg- = i^; 

h — a — §' {See Note^ page ^2.) 

N 



A = 



2 P sin a 



A'^ ^ 



cos /?' 

a=z _ or = a' =:d ■{■ 2 a 

P ;r 

2 a =i 2 cos o' 

d! for gear = <^ for pinion. 

a for pinion z= b for gear. 

Note. — See Foot Note on page 13. 



PROVIDENCE, R. I. 



17 




l8 BROWN & SHARPE MFG. CO. 



The formulas given for a^ and a^ (when C, N„ and N,, are 
known) undergo some modifications for values of C greater 
than 90°. 

P'or bevel gears at any angle but 90° we may distinguish 
four cases ; C, N^, Nj, being given. 

/. Case. See pages 14 and 16. 

//. Case. C is greater than 90°. 

sin (180 — C) ^ sin (180 — C 
tan a^-=z 5^ ^ ; tan a^ = — ?^ 

_^-cos (180-C) S-"~^^^ (i8o-C) 

III. Case, a^ = 90° ; tar^, = C — 90° 

IV. Case. 

sin E ^ sin E 

tan a^ = — - ; tan aj, 



cos E - ^^ ^« - cos E 

For an example to apply to Case III., the following condi- 
tion must be fulfilled : 

N„ sin (C - 90°) = N, 

To distinguish whether a given example belongs to Case II. 
or case IV., we are guided by the following condition : 

T ivT • /'n °\ J smaller than N{„ we have Case II. 

IS : rM„ sm (u - 90 j -j ^^^^^^ ^^^^ ^^^ ^^ ^^^^ ^^^^ ^^ 



PROVIDENCE, R. I. 19 



UNDERCUT IN BEVEL GEARS. 

By undercut in gears is understood a special formation of 
the tooth, which may be explained by saying that the elements 
of the tooth below the pitch line are nearer the center line of 
the tooth than those on the pitch line. Such a tooth outline is 
to be found only in gears with few teeth. In a pair of bevel 
gears where the pinion is low-numbered and the ratio high, we 
are apt to have undercut. For a pair of running gears this 
condition presents no objection. Should, however, these gears 
be intended as patterns to cast from, they would be found use- 
less, from the fact that they would not draw out of the sand. 
We have stated on page 2 (see Fig. i) that the base of our 
involute system is the 14)^° pressure angle. If a pair of bevel 
gears with teeth constructed on this basis have undercut, we 
can nearly eliminate the undercut — and for the practical work- 
ing tliis is quite sufficient— by taking as a basis for the con- 
struction of the tooth outline a pressure angle of 20°. 

The question now is : When do we, and when do we not 
have undercut ? Let there be : 

N = number of teeth in gear. 
n = number of teeth in pinion. 



N 
where we have undercut for/ less than 30. 

This formula is strictly correct for epicycloidal gears only. 
It is, however, used as a safe and efficient approximation for 
the involute system. 



20 



BROWN & SHARPE MFG. CO. 



DIAMETER INCREMENT. 



Rule. — The ratio being given or determined, to find the outside diameter 
divide figures given in table for large and small gear by pitch (P; and add 
quotient to pitch diameter. 







GEARS. 1 






GEARS. 




GEARS. 


RATT*^ 






RATT*^ 






RATIO . 








Large 


Small 






Large 


Small 




Large 


Small 


1.00 


1:1 


1.41 


1.41 


1.65 




1.05 


1.70 


4.40 




.45 


1.94 


1.05 




1.37 


1.42 


1.67 


5:3 


1.03 


1.72 


4 50 


9:2 


.44 


1.95 


1.07 




1.36 


1.43 


1.70 




1.01 


1.73 


4.60 




.42 


1.95 


1.10 




1.35 


1.44 


1.75 


7:4 


.99 


1.74 


4.80 




.41 


1.96 


1.11 


10:9 


1.34 


1.46 


1.80 


9:5 


.97 


1.75 


5.00 


5:1 


.39 


1.96 


1.13 




1.33 


1 46 


1.85 




.95 


1.76 


5.20 




.38 


1.96 


1.18 


9:8 


1 33 


1.47 


1.90 




.93 


1 77 


5.40 




.37 


1.96 


1.14 


8:7 


1.32 


1.49 


1 95 




.91 


1.78 


5.60 




.36 


1.97 


1.15 




1.31 


1.50 


2 00 


2:1 


.89 


1.79 


5.80 




.34 


1.97 


1.16 




1.30 


1.51 


2 10 




.87 


1.80 


6 00 


6:1 


.33 


1.97 


1.17 


7:6 


1.30 


1.52 


2.20 




.84 


1.81 


6.20 




.32 


1.97 


1.18 




1.29 


1.53 


2 25 


9:4 


.82 


1.82 


6 40 




.31 


1.97 


1.19 




1.28 


1.53 


2.30 




.80 


1.83 


6.60 




.30 


1 97 


1.20 


6:5 


1.28 


1.54 


2.33 


7:3 


.78 


1.84 


6.80 




.29 


1,98 


1.23 




1.27 


1.55 


2.40 




.76 


1.85 


7 00 


7:1 


.28 


1.98 


1.25 


5:4 


1.25 


1.56 


2.50 


5:2 


.75 


1.86 


7.20 




.27 


1.98 


1.27 




1.25 


1.57 


2.60 




.73 


1.86 


7.40 




.27 


1 98 


1.29 


9:7 


1.24 


1.58 


2.67 


8:3 


.71 


1.87 


7.60 




.26 


1 98 


1.30 




1.22 


1.59 


2.70 




.69 


1.87 


7 80 




.26 


1.98 


1.33 


4:3 


1.20 


1.60 


2.80 




.67 


1.88 


8 00 


8:1 


.25 


1.98 


1.35 




1.18 


1.61 


2.90 




.65 


1.89 


8.20 




.24 


1.98 


1 37 




1.17 


1.61 


3.00 


3.1 


.63 


1.91 


8 40 




.24 


1.98 


1.40 


7:5 


1.16 


1.62 


3.20 




.60 


1.92 


8.60 




.23 


1.98 


1.43 


10:7 


1.15 


1.63 


3.33 




.58 


1.92 


8.80 




.23 


1 98 


1.45 




1.13 


1.65 


3.40 




.56 


1 92 


9.00 


9:1 


.22 


1.99 


1.50 


3:2 


1.11 


1 66 


3.50 


7:2 


.54 


1.93 


9.20 




.22 


1.99 


1.53 




1.10 


1.67 


3.60 




.52 


1 93 


9.40 




.21 


1.99 


1 55 




1.09 


1.67 


3.80 




.50 


1.94 


9.60 




.21 


2.00 


1.58 




1.08 


1.68 


4.00 


4:1 


.49 


1.94 


9.80 




.20 


2.00 


1.60 


8:5 


1.07 


1.68 


4.20 




.47 


1.94 


10.00 


10:1 


.20 


2 00 



Note. — To be used only for bevel gears with axes at right angle. 



PROVIDENCE, R. I. 21 



TABLES FOR ANGLES OF EDGE AND ANGLES 

OF FACE. 

The following three tables have been computed for the 
convenience in calculating datas for bevel gears with axes at 
right angle. They do not hold good for bevel gears with axes 
at any other angle. 

To use the tables the number of teeth in gear and pinion 
must be known. 

Having located the number of teeth in the gear on the 
horizontal line of figures at the top of the table, and the num- 
ber of teeth in the pinion on the vertical line of figures on the 
left-hand side, we follow the two columns to the square formed 
by their intersections. 

The two angles found in the same square are the respective 
angles for gear and pinion. The tables are so arranged that 
the angle belonging to the gear is always placed above the 
angle for the pinion. 



22 



BROWN & SHARPE MFG. CO. 



TABLE I 
Angle of Edge. 





41 


40 


39 


38 


37| 


36 


35 


34 


33 


32 


31 


30 


29 


28 


[27] 


12 


7341 

16*19 


73*18 


72^54 
I7'6' 


72-28 
17 V 


72*2 

17*58 


7I°3+ 


71*5- 

18*S5 


70*34 
19*26 


70*1' 

19*59' 


69*26 
20*34 


68*so' 
21 'o' 


68*.2' 
21*48- 


67*31 

22*29 


66*48 

23*12 


66*2- 

23*58 


13 


72'js 
17 V 


71 59 
IB'V 


71*54 
18*26 


71*7 
18*53' 


70*39 
I9*2l' 


70" 9' 
I9°s.' 


59*37' 

20*23' 


69*5 

20*55 


68*30 

21*30' 


67*53 

22*7' 


67*15 

22*45 


66*34 
23*26 


65''s.' 

24*9 


65*6' 

24*54 


64'.7' 
25*43' 


14 


71 V 
18 V 


7043 
19*17' 


70*15 

1945' 


6946 

8o*m: 


69*16 

20*44 


68*45 

e 1 'is 


68'ie' 

2\'48 


67V 

22*23 


67'o' 
23*0' 


66*23 

23*37 


65*42 
24*18' 


64*59 

25* r 


64*14 

25*46 


63*26 

26*34 


62*86 

27*24 


15 


69S4: 
80*6' 


68*26 
20*3* 


68"se 

21*2" 


6888 
21*32 


67*56 

22*4 


67V 
22*37 


66*48 

23*12' 


66*12 
23*4« 


65'33' 

24°27 


64'b' 
85*7' 


64*,o' 
25*50 


63*26 

26*34- 


62*19 

27*21 


fel*48 
28*11' 


60*5* 
29*3 


16 


68V 


68*«' 


67V 

ezV 


67*,o 
22*50 


66*37 
23*23' 


66*2' 
23°S8 


65*26 
24*34 


64*48 

25*12 


64-8 

25*52 


63*26 
86*a4 


62*42- 
27*18 


61*56 

28°4- 


61*7- 
28*si 


60^15 

29*45 


59*21' 
30*39' 


17 


6729' 


66S8' 
23*2 


66-27 

23*33 


65*54 

2**6 


65*19 
24*4.' 


64*43 

25°.7' 


25*S4 


63*26 

26*34 


62*45 

27°.s 


62*1' 

27*59 


61 "15- 

28*45 


60'28 
29°32 


59*37 

30*« 


58*44 

31*16 


57*48' 

32*12 


18 


23*48 


6546 

24°i4 


65-kV 
24%fe 


64*S9 
25*2. 


64^4' 
25*56 


63'fe6 
26*3^ 


62V 
27° a 


62' fe 
87*5^ 


ei'o 

28*37' 


60*38 

29*K 


59*5. 
30*9 


59*2 
30*58 


58*10 

3i*M 


57V 

32*44 


56*19 
33*41' 


19 


65-8 
2Vn 


2S*^4 


64°e' 

es*s8 


63*26 

26*84' 


62*49 

27*11 


62*10 
27*50 


61*30' 
28*30 


60*48 
29*K> 


60*4 
29°S6 


59*18 
30*4i 


58*30 

31*30 


57 » 
32*ei 


564* 
33*14 


SB'si' 
34*9' 


54*52 
35*8' 


20 


64-0- 
26*0 


63*36 

26*34- 


62*51' 
27*9' 


62*,4 

27°46' 


61 V 

28*23 


eo'svi 

29'* 


60'.5 

29*45 


59*32' 

30*28 


5847' 
31*13' 


58*0' 
32*0' 


57w 
32*50 


56' ,9 
33*41 


55*« 

34*36 


54^28 
35*32' 


53'w 

36*« 


21 


62-s. 

27*7' 


62*« 
27°4t 


6lV 
28*(e' 


61*4' 

28°re 


eo^Es 

29*3S 


53''45' 

30*.s' 


30*58' 


58m 

31*42 


57V 

32*28 


56*43' 

33*17 


55m' 
34*7- 


55"o- 
35°o' 


54*5- 
35*55 


53*7 

36*53' 


52*8 
37«- 


22 


6lV 

28°3 


ei°»' 

28*49 


60°34 
29°8fe 


59S6 
30*4' 


59*15' 

30*45 


58*34 
31°26 


57'5.' 
32*9' 


57*6' 
32*si 


56*19 
33*4." 


55*29 

34*31 


54*38 
35*22 


53*45 

36*15' 


52*43- 
37*11' 


51-50' 
38*10 


50*49' 

39*11' 


23 


60%i 
29 18 


60*6 
29°5i 


59'28 
30*32 


58*49 

31*1.' 


seS 

3r52 


57*25 

32*35 


56*41' 
33*19 


555S' 
34*s' 


55S' 

34*53 


54-18 
35°4i 


53*26 

36**i: 


52*31' 
37*2^ 


5135 

38» 


50*3,; 

39*24 


4^^i 

40*86 


24 


S9'39 

30\/ 


59'2- 

30*58 


58n 

31*37 


57*44 

32*16 


57"8 
32*58 


56*19 

33*41' 


55m 

34*27' 


54*47' 

35*3 


53*S^ 
36*2 


53*7' 
36*53' 


52*15 

37*45 


5 l-zo' 
38*40' 


50"23' 

39*37 


4324 
40*36 


48*22 

41m' 


25 


S8*S8 


58''q' 
32'o' 


5720 
324«; 


56*40 
33*Eo' 


55^57 

34*3' 


55-13' 
34*47' 


54*28- 
35V 


53°4o' 
36*20 


52*5. 
37*9' 


52*o' 
SS'o' 


51"?- 
38°sa- 


SO'iz 
39*48 


49-«. 

40*46 


48^14 
41*46 


47*12- 
42*48 


26 


5737' 

3Zzi 


56 58 

33*e' 


56' 19 
33*41 


5537 

34*23' 


54-54- 
35*6 


54-10 
35*50 


53'24 
36*36 


5236 

37V 


51-46 
38*14' 


50S4 

39°e' 


50%' 
39°S9' 


49*5 
40*55 


48S 
41*53 


47%- 

42*53' 


46*5 
43*55 


27 


56-38 
33*a 


55-5, 
34.° i' 


55**8 

3442 


54V 
35*M 


53*53 
36*7 


53"7' 
36°53 


52^21 
37*39' 


SIM 


5O43' 
33*17' 


49"5l 
40*9 


48"s7 

41*3 


4€'^o' 
42*0- 


47V 

42%7 


46''z 
43*58 


4.S* 


28 


SS4*; 

34V 


55*0 
35*0' 


54*19 
35*41 


53*37 
36*2i 


52*53 
37*7 


52^8 
37^ 


51*20 
38*40 


50*32 
39*28 


49»4.' 
40*19 


48*46 
41*12 


47-55 
42*5 


46*58 
43*2 


46*0 
44*0' 


45* 




29 


54V 
35*6 


54S 

35*57 


53a 

36*38 


52*33 

37*ei' 


51*55 
38* S 


51-9- 
38*51 


5021- 

39*39' 


43*32 
40*28 


48*4.' 

41*19 


47*4? 
42*11' 


46'» 

43*6' 


4518' 

44*2' 


45* 






30 


53*48 

36*iz' 


53*7' 

36*53 


S2*t6 
37*34 


51*42 


50*58' 
39°t' 


50-.2 
394^ 


49*24 
40*36' 


48^35 
41*25 


47*43 

42*17 


46*51 
43*9 


45*56 
44*4' 


^5' 






31 


52*54 

37V 


5e*.3 

37*47 


5l°3,' 
3*89 


50*48' 
39*2 


50*2' 
39w 


49*ife 

40*44 


48*2«' 
41*32 


47°39 
42*21 


46*47 
43*13 


45*54 
44*6 


45* 






32 


52*2' 
37°s« 


51*20 

3840 


50-38 
39*22 


49*54 
40*6 


49*9 
40*si' 


48'« 
4l'3»' 


47^34 
42*26' 


46*44 

43*16 


45n 
44*7 


45* 






33 


Sl'io' 
38*50 


5029 

39°3i 


49''46 

40*4 


49'2- 

40*56 


48*.6- 

41*44 


47*29 
42*2. 


46*'4. 
43°.i» 


45"5. 
44*9 


45" 






34 


50W 
394o' 


48*38 
40*28 


48^55' 
41*5' 


48'h' 

4I*4S' 


47*25 
42*35 


46*38 

43*22 


45^0 
44*10 


45* 






35 


49^3.' 
40»' 


4848 

4(*ie 


48*5 
41*55 


Al'v 

42*39 


46-35 
43*25 


45*48 

44*,2 


45* 






36 


48*41 

41*17 


48' 
42*0 


47''(7' 
42%i 


46*33 

43*27 


45'47 
44*13 


45* 






37 


47"5* 
42*4' 


47*,4 
42*4< 


4<5''30 
43*30 


45i* 
44*14 


45° 






38 


4r.o 
42*so 


46'a 
43*31 


4^45 

44*15 


45* 






39 


46> 
43V» 


45*43 

44'? 


45' 






40 


454a 
44*18 


45' 






41 


45' 













PROVIDENCE, R. I. 



23 



TABLE I. — {Continued^ 



Angle of Edge. 





26 


25 


24 


23 


22 


21 


20 


19 


18 


17 


16 


15 


14 


13 


12 




12 


2446 


64'22 
25°38 


63*26 
26*34 


62^27 
27°33 


61 Vs 

28*37' 


60'i5 
29°4S 


59*2 

30*58 


574* 

32*16 


56 "i9 
33°4i 


54 47 

35*13 


53% 

36*53 


51 20 
38*40 


49*24 
40°36 


47*17 
42*43 


45* 




13 


26*34' 


62-3. 

2729 


61*33 
28*27 


SO'ji 

2^°29 


59*25 
30*35 


5e'v4 

3 1 °46 


56-58 
33* 2' 


55-37 

34*23 


54-10 
35*50 


52°36 
37*24 


50*54 
^9*6 


49-s 
40°55' 


47*7 
42*5i 


45* 






14 


61 42 
2 8° 8 


604S 
29° 5 


59°4S 
30*15 


58*0 


57V 
32*28 


56*19 
33*41 


55*0 
35-0 


53-37 
36-2i 


52*8 

37°52 


50m 

39*28 


4848 

4ri2 


46''58 

43'e' 


45' 






15 


60S- 
29°53 


59- 2 
30*58 


58 
32*0 


56^> 
33*7 


55« 
34"i7' 


54-28 
35*32 


53'? 

36*53' 


51 V 
38°i8 


5012 
38*48' 


48 'iS 

41*25 


46''5I 
43°9 


45° 






16 


58'z} 
3lV 


57 "23 
32°37' 


5fc°.9' 
33*41 


55*11 

34°43 


53*58' 

36*5' 


52*42' 

37*18' 


51 « 
3840 


49-54 

40*6 


48*22 

4l°3e 


46'44' 

43*16 


45* 






17 


56V 
33*11' 


5547 

34°i3' 


5441 
35*19 


53°32 

36*28' 


52- .8 

37*42 


51-0- 
38° 0- 


49'36- 
40°22 


48-M 

4143 


4€-38 

43*22 


45* 






18 


55*18 
34V 


54'.5' 
354S 


53°7 

36*53 


5.V 
38*3' 


5043' 

39*17 


49'^ 

40*36 


48*0 
42' g- 


46"» 
43*27 


^s° 






19 


53S. 
36*9' 


37 V 


5I°38 

SSk' 


39V 


49*11' 
4049' 


47*52 
4?° 8' 


46-28 

43*52 


^C 






20 


37Vv 


51-20 
3840 


50V 
3948 


48*59 

41°.' 


4743' 
42*17' 


4€''24 
43*36' 


45- 






21 


51^' 

aa'se' 


49''58 
40' 2' 


4SV 
4['« 


47*3^ 

42*64 


46*20' 
43°40 


45° 






22 


4946 

40' 14 


48V 


47-29 
42*31 


46*ife 
43-44 


4-5° 






23 


48*'3o 
4r30 


47*23 
42*37' 


4€"«- 

4347 


^5° 






24 


47 '17 

42'43 


46*1 
43°5o' 


4-5° 






25 


46S' 

43*53 


45' 






26 


4-5* 







tan a^ = 



tan a,, = 



Na 



No 
{See page ij.) 



24 



BROWN & SHARPE MFG. CO. 

TABLE 2. 

Angle of Edge. 





72 


71 


70 


89 


88 


67 


fifi 


65 


64 


63 


62 


61 


60 


59 


58 


57 


12 


80°8i 
9'ti 


80» 
9*3S 


80« 

9°4» 


8e*» 

a*s2 


79-» 

10* V 


79*5t 
10*9 


79*42 
lO'ia 


79*32 
»o*ae 


79;ri 

10*37 


79*,3 

10*47 


79^3' 

.0*57 


78*5i 

ii°e' 


I.°I9 


78m 
11*30 


78*19 

1 .°4I 


It S3 


13 


79*46 


79*37 
10*23 


79«; 

I0°3l' 


79*« 
.0*40 


79*,|- 
10*49 


79*,' 

lo'ss 


18*51 
.1*9' 


7«^4. 
11*19 


78''3l' 
..*29 


78*20 

11*40 


78*9 
1 1*51 


77*58 
12*2 


77*46 
.2*14 


77*34 
12*26 


77*22 
12*38 


77*9 
.2'5l 


14 


79-0 


78*51 
11*9 


78%; 

11*19 


76*32' 

U*2e 


78*22 
11*38 


7e°.i 

11*49 


78*1' 
11*59 


77*51- 
12*9 


77*40 
12*20 


77''28 
12*32 


77*.7 
.2-4i 


77*5 
12*55 


76'si 
13*8 


76-39 
13*2.- 


76*26 

13*3i 


76*12 
13*48 


15 


78V 


78*V 

11*56 


77*5.' 
12*6 


77*44 
12*16 


77*3* 

12*26 


77*2i 

12*37 


77*.e 

12*48 


77*o- 
f3*o' 


76*48 
13*12 


76*3i 
I3*»i 


76*e4 

13*3^ 


76*,i' 
13*49 


75*58 
14*2 


7S*4i 
14*1^ 


7S*3i 
14 30 


7S*.5 
14*45 


16 


77'28 


77*i 

12*42 


77*7' 
12*63 


76°S7 
13*3' 


76*45 
13*15 


76°3i 

13*26 


76*22 
13*3^ 


76*.6 
13*50 


75*58 
14*2 


75*45 
14*15 


75*32 
14*28 


75*18 
14*42 


75V 
H*56 


74*49 

15*11' 


74*35 

15*2^ 


74-19 
15*41 


17 


7643 


7^» 
I3tt 


76*2i' 
13*39 


76*10 
13*50 


75*58 
14*2 


75°4« 

14* (4 


75*33 
14*27 


75°a.' 
14*M 


75*8' 
14*52 


74*54. 
15*6 


74°4(i 
15*20 


74*25 
15*35 


74*11 
15*49 


73*56 
16*4 


73*40 
16*20 


73^24 


18 


75^58 
14° t 


75^6 

14*14 


75-3S 
I4*2S 


75*21 
14*37 


7S°,6 
14*50 


74°5ft 
IS°2 


74*45 
15*15 


74*3. 
15*29 


74*17 

15*43 


74*3 
»5*5» 


73*49 

16*11 


73*33 
16*27 


73*18 

I6*4i 


73*2 
16*58 


72*45 

n*i5' 


72*2i 

17*31 


19 


7^o 

»4%7 


75* I- 

I4°59 


7449 

15*11 


74*36 
15*24 


74*23 
15*37 


1550 


73*56 

16*4 


73*42 
16*18 


73*2i 
16*32 


73*23 
)6*4l 


72*58 

17*2 


72*42 
17*18 


72*26 
17*34 


72*9 

i7°si' 


71*52 
18*8 


71*3* 
.8*26 


20 


74-« 


74'>6 

15*44 


74*3' 
15*57 


73^50 
16*10 


73*37 

16*23 


73*23 
16*42 


73*9 
16*51 


72*54 
17*6 


72*39 
17*21 


72;2i 

17*37 


72*7' 
17*53 


71*5.' 

se'a' 


71-34 
18*26 


71-16 

18*44 


70*59 

19* r 


7Cf46 
19*20 


21 


73-45 

i«ri5 


7^*32- 

I6°2a 


73*.B 

16*42 


73*4- 
16*56 


72;so 
17*10 


72*36 
I7*a4 


72*2. 

17*39 


72*6 

17*5* 


7.^50 

18*10 


71*34 
18*25 


7.*17 

18*43 


7.*0' 
l9*o' 


70*43 
19*17 


70-24 
19*36 


70*6- 

19*54 


69*46 
20*24 


22 


73- .• 
16*59 


72-47 
i7°3 


72*« 
17*27 


72^9 
17*41 


72*4 
I7°S6 


71*49 

18*11 


71*34 


7.*i6 

18*42 


71*2 
18*58 


70*45 
.9*,5 


70*28 
19*34 


70*10 
19*50 


69*52 
20*8 


69*33 
20*27 


69-13 
20*47 


68*« 
21*6 


23 


7Z*F7' 

r7%3 


78*3 
I7*S7 


71*49 

I8°u' 


7lV 

18*26 


7I*N 
I8°4l' 


71*3 
18*57 


7oV 
I9«« 


70*90 

19*»i 


70*14 
.9*46 


69*57 
20*3 


69;39 
20*21 


69*20 
20*4d 


69*e 
20"S8 


68*42 
21*18 


68*22 
21*38 


68*2 

21*58 


24 


7r3» 

18 a; 


7I*« 
I8°4i' 


71*5 
18*55- 


7tf4, 

19*..' 


70*34 
19*26 


70° 17 

19*43' 


70*,' 

19*59 


89*4i 
20*16 


69*2^ 
20*3^ 


69*9 
20*5. 


68*50 
2l*.(i 


68*3.' 

21*29 


68*12 
21*48 


67*52 

22*8 


67*31 
22*29' 


67*,o 
22*50 


25 


70*51 
J9*9' 


70*36 

19m' 


70*ti 
19*39 


70*5' 
I9°5S 


69*49 


69^2 
20°2« 


69*»5 
20*4S 


68*57- 
21*3' 


6840 

21*20 


68*2.- 
21*39 


68*3- 
21*57' 


67*43 
22*17 


67*23 
22*37 


67*2' 
22*58 


66*4.- 
23*19 


66*19 
23*41 


26 


70*9 
13*51 


20*7 


69*37 
20*23 


69*2.' 
20*39 


ea*4 

2056 


68*48 
21*12 


68^30 
21*30 


68''.2 
2.*46 


67*5^ 
22*6 


67*34 
22*26 


67*15 
22*45 


66*55 
23*5 


66-34 
23*26 


66*.3 

23*47 


65*s. 
24*9 


65*29 
24*3.' 


27 


W*27 

20*SJ 


69-.0 
2O*S0 


68*5* 
2.*6 


68*38 
21*22' 


66*«; 

21*40 


68*3 

21*57 


67*45 
22*15 


67''2*- 
22*M 


67*8 
22*52 


66*46- 

23*12- 


66-28 
23*K 


66*7' 
23*53 


65*46 

24*14 


65*25 
24*35 


65*2' 
24*58 


64*39 
25*2.' 


28 


e8-4s' 

2I*» 


68-29 
21*31 


68*.2 
21*48 


67*55 
22*5 


67V 

22a 


6r.9 

2241 


67* r 

22*» 


66°42 

23*« 


66*22 

23*38 


66*2 
23*54 


65*4i 
24*18 


24*39 


64*59 

25* r 


64*37' 
25*23 


64*K1- 

25*46' 


63*s6 
26*i6 


29 


68V 
21*54 


«47 

22*13 


6^0 
22*se 


e7:,2 

2246 


66*54 

Z3*6 


66*3i 
23*24 


66*,7' 
23*43 


6S'57 
24*3 


6?37 

a4-2i 


65*16 

24*44 


64*55 
25*5 


64**i 
25*26 


64*12 
25-48 


63*50 
26*10 


6^26 
26*34 


63*2 
26*58 


30 


22W 


22*W 


6b°46 

23*12 


66*30 
23*3« 


23°48 


6^S2 

24*9 


65*30 
24*27 


6SV 
24*4fe 


G4.*« 
25*7 


64Si 
25*28 


64*10 
25*50 


€3*49 

26*n' 


63*26 
26*34 


63-3- 
26*57 


62*39 
27*z( 


62*14 
27*4« 


3i 


23*18 


6(.'» 


66V 

23*54 


65*48 
24*12 


65*« 
24*3. 


e5*.i 

24*t» 


64*50 
25*«i 


64*30 
25*30 


64.*9 
25-S. 


63*48 

26*«; 


63>6 
263^ 


63'^3 

26*5^' 


62*46 
27*20 


62*18 
27*42 


6»-B3i 

28*7 


61*28 
28*32 


32 


66V 


65*44 

24*16 


6S-26 

24*3^ 


6S-7 
24*53 


&)*4e 
25V 


64*26 
25*32 


25S2 


6ly*47 

26*13 


63*26 
26*34 


6^4 
26*56 


62*42 

27*« 


62*19 
27*4.' 


6.;56 

284 


6.-,2- 
28*28 


61*7 
28*53 


60*41 
29*19 


33 


65°d 

24*37 


2456 


64*45 

25*15 


64*26 
25* J4 


64*7- 
25*53 


63*47 
26*13 


63-26 

26*34 


2655 


6243 
27*17 


62*2i 
27*3^ 


6.-58 
28*2.' 


6l°3i 
28*25 


€l*.i' 
28*49 


60*47 

29*13 


6(^2,' 
29*39 


30 4- 


34 


W'43 
i5*.7 


fc4*2S 
25*35 


64*.' 

25*55 


63>- 
26**4 


6326 
26*34 


^3> 

26 55 


6245 

27 15- 


62*23 

27*37 


62:.' 

27 59 


61*38 
28*21 


6.*,5 
28*45 


60*52 

29* e' 


29 32 


60*3 
29*57 


59*37 
30*23 


59:.i; 

30*49 


35 


^5*55 


63%5 
'/6*is' 


63*26 

26°>V 


63; 6 
26* S4 


624b 
27*14 


62''25 

2 1* si 


62*^' 
27*56 


6.*4i 
28*18 


28*41 


60*57 
29*3 


60*33 
29*27 


60*9 
29*51 


59>5 
30*15 


59;.9' 
30*41 


58*53 

31*7 


58*27 

3.*3J 


36 


63*26 
26*M 


63*7 
26*53 


62V 
Z7*I3 


62*27- 
27*33 


62*6 

27*5^ 


6i*45 
28*«5 


6J*23' 
28*37 


6.*,' 
28°!» 


60*33 
29*22 


60*,s 
29°4i 


59*51 
30-9 


59*27 
30*33- 


59*e 

30*58 


58*37' 
3.*23 


58*16 
31*50 


57*43 

32*n 


37 


62-4e- 
a7*.2 


62*2i 
27*32 


62*8 

27a 


6l*4i 
28'« 


6.*27 
28°33 


61° b' 
28>i 


60*44 

29*16 


eo\.: 

2939 


59*5i 
30*2 


S9°»s 
30*2i 


59*10 
3O*S0 


58*46 
3.-|4 


58*20 
31*40 


57*54 
32*6 


57*28 
32*32 


57*.' 
32*5> 


38 


27*'» 


6l°Si 
28°»' 


61*30 

28»<i 


28*51' 


60*4i 
29*ii 


6026 

29**; 


60-^ 
29*56 


5941 
30*19 


59^8 
304^ 


58-64 
3.°6 


58*^6 

31*3(i 


58*S' 

31*55 


57*3,' 
32*21 


57*.3' 

32*47 


56^46 
33*14 


56*« 
^3-4i' 


39 


6l*:»' 

28*27 


6.*li 
28*47 


60*53 
29*7 


60*31 
29» 


60*10 
29*50 


59*48 

so'«i 


59-25 
30*35 


59^ 
30*5* 


58*39 

3.*2i 


58°W 

3l°A6 


57*^ 
32*10 


57*24 
32*3^ 


56*58 
33*2 


56*32 

33*28 


5^6 
33'W 


55*87 
34*23 


40 


6o>,; 

293 


60''s6 
29*4.' 


60'is 
2945' 


59*53 
30*7- 


59*32 
30*28 


59*.0 
30*5i 


5^47- 
3.*.3 


58'« 

31*3^ 


58*0' 
32* o' 


57*35 

32*2^ 


S7°.ri 
32*50 


56*44 

33*16 


56*19 

33*4.- 


SS*S2 
34*8 


55*24 

34*35 


54*57' 
35*3 


41 


60^6 


60° o' 
30*0 


59;3S 
30*21 


59- .7- 
30*43 


58*SS 


58*32 
3l'28 


SB's 
31*51 


57*4^ 
32*5 


57*2.- 
32*39 


56*57 
33*3 


56-3Z 

33*26 


56*6 
33*54: 


55*39 

34*2 1 


55°.i. 
34*48 


54*44 

35''.6 


54*16 
35*44 


^ 


59*-« 
30*is 


S9-*i 
30*36 


59*3 
30*57 


5e''4« 

31*20 


50*,e 

3.*4i 


57*SS 
32*5 


57^2 
32*28 


57'8 
32*52 


56*43 
33*17 


56*.9' 
33*41 


55*53 
34*7 


55*27 
34*33 


55*0- 


54*33 

35*27 


54-5 
35« 


53;37 
36*23 



PROVIDENCE, R. I. 



25 



TABLE 2.— {Continued) 
Angle of Edge. 



56 



5554 



5352 



51 



50494847 



4645 



44 



43 42 



12 



77 
12' 6 



54 77 



42 77^28 
19 12*32 



77^15 
I2%5 



77V 



7^46 



7^ 



76-14. 
1346 



7558 



7S4l' 



7^2a l^\ 



TA-AS 



74°25 74 a 



i^Be 7^4 
•2 14' 



14*56 IS'lS 

73*61 7i'K 
16*7' I6°2a 

72^7^ 



.35 IS 

iV I7°ig' 
^ 7I''3* 



13 
li 

15 
]6 
j7 

J8 

19 
20 

21 
22 
23 
24 
25 
26 
27 
28 
29 
30 
31 

^ 
33 

34 
35 
36 
37 
38 
39 
40 



76s6 
13* 



76° 
I3%il 



76'l3 
1347 



7S'^|75*2B 
H'iV |4°32 



^56 74*39 

i*4.' JS'ai 



7S°e 

14*52 



7451 
IS°9' 



74»a 
15*28 



74 13 
»S*47 



7S'se 

14*2 



75V 
14*48 



74*21 
IS"39 



74'3 
(B*B7 



73^73 



es 

16**5 



73"'4 

ie*5«{ 



17 17 



17*3^ 



18*2 1 le'W 

'1*34 71° »' 7O°4i70V 

*2b »°S0 19° 14 »9*33 



75o 
1 5*0 



74*44 74*29 
15*16 15*31 



74 12 
I5'4» 



ijss 73* 



73 18 
16*42 



72 S3 

»7*r 



72 391 
J7'2t 



72 18 

»7*42 



7|''S6 



7^26 70"! 
19*34 >9*59 



74-3 
.5'57 



7347 7330 

16*13 le'ai 



73 la 
I6''4e 



72°S4 72°35 



71 S5 



71 

ie''26| 



71 12 
18*48 



70°49 
I9'll 



69*35 63^9 
20V 205» 
68*86 67V( 



73'7 
16*53 



•7 II 



17 29 



72 i3 
17*47 



18 6 



7 (''34 
I8°2fe 



7l''l3 
1 9*47 



7052 

i9°e 



70*^0 70V 



6943 
20"i7 



69*17 68'S2 
2043 a 1*8 
68 12 67*45 
21*48 22*15 
67*6' 66*38 



l*3» 22*z' 
67*17 66*48 
22*43 23*12* 

65*39 



721 
17*49 



7l*53|7r3» 
18*7 



18 26 



7I°I5 
18*4$ 



70V 70''jj 



19 6 



19 27 



7012 
»9' 



69 so 
20*10 



6926 

26 



69^3 
20*57 



68 36 
21*22 



70*57 70*37 



71 15 
18*45 



19 I 



19 23 



70* r 69 
19*S9 20*19 



7017 
19*43 



69*5^ 69*34 



20 



20 W 



68 25 

21*35 



67 59 
22 



67^34 

22*26 22*541 23*22 



23*50 24*21' 



I9*3« 



» 68*45 
2Qr»V 2»'l5 



6919 
20*41* 



157 68 35 
*3' 21*25 



68^12 

21*48 



€748 

22*12 



6723 
22*97 



6^57 
23*3 



66 
23*^301 



'36 6€f 



6^33 65 



2 

23*58| 24' 
64*59 64*29 
2S*r 25*31 
63*57' 63* tb 



24*57 25*28 
63*58 63*26 
26* g 2^3» 
62*54 62*21 
27*6 27*39 
6i*sz 6l*t« 



6»M 69'*o 



6823 
21*37 

67^ 
22''3J 



66"o' 67*37 
22*23 



67*13 
22*47 



66*48 
23*12 



662* 65 S5 



6528 
24*32 



6833 
21*27 



6812 
21*48 



67°50| 
22*10 



67''4 



66*4<> 



22 56 23 20 



6615 
23*45 



6549 

24*11' 



65^23 

24*37 



64ss 
25*5 



64 26 
25*34 



26*3 
62*56 



26 34 

62 

27''3«l28'e 



67*41 
22*19 



67*18 
22*42 



66 5SI 
23*5 



6?5 
23*28 



66" 8 
23*S2 



65*44 

24*16 



65 18 

2442 



6451 
25*9 



64*24 63 55 
2S*36 26* S 



26 >t 



2842 
60*S« 60^is' 
29*10 2945 
59*50 59"»4: 



6646 

23*12 



66*26 66*2 



65*38 
24*22 



6S"i4 

24*46 



64°4e 

25*12 



6422 
25" 



63*s* 
26*6 



63 26 

26*34 



62*57 
27*3 



6227 
27*33 



61*56161*23 
28*37 



6?57 

24*3 



6533 
24*27 



6^ 
25*18 



65 9 
24*51 



6445 
25*15 



64 20 
E5*4o' 



63*53 

26*7 



63 26 
26*34 



62 s« 
27*2 



27 31 



6I» 
28*1 



6I''2^ 

28*31 



60 57 
29*3 



29j 
5S' 
30* 

3£s8 3l^ 
58*7 57°3i 
31*53 32''2» 



3c30 



lO l 3046 

58*50 Se°i4 
si' id 31*46 
S7*S3S7^ 
32*7 32*44 
56*5iS?W 
.33*4 33*4l' 
^57 55^ 



656 

24*541 



r6<« 

2^4i 



63*152 
26*8 



63*26 



ei'ss 



26 34 27 I 



6231 
27*29 



623 

27*57 



6133 
28*27 



61 3 
28*57 



60*31 
29*29 30 I 



64'i6 
25*44 



63*5.' 
26*9 



63*2; 



62*34' 
27*26 



62:^' 
27* 



^36| 

27 



M28 22 



61 8 
28*52 



60^ 



29 ta 29 53 



3025 



63*26 

26*34 



63* i' 
26*9» 



6?? 
27*51* 



61*42 61*14 



2818 



26" 



6O45 
29*15 



iSS 



59^ 



2946 30 IS 



62*37- 
27*23 



6^12 



61*45 



27*46 28*15 



61*19 
28*4i' 



60 51 
29*9 



60"2J 

29*37 



W7 

29*59 30°28J3O 



59S3|5923 
30*7 



5862 
31*8 



59 «3 
3047 
58*19 

31*41 



5?4» 

31*20 

5r46 
32*14; 



57 12 



56*37 



3249 33 23 

56*19 55 



3A'*o 34*87* 
55^5?5 
34*CS 35*32 
54*12 53*34 
35*46 36*g» 
53*21 52*42 
36°W 37*18 
52*29 Sl'so 



28*11' 



bfti 60 
2837 29*36 



6C^ 60°c 
29*2* 29*54 



60"29 

2»'3i' 



S9 2 



58 
32' 0' 



57*27 

32*33 



33*7 



33ai 



34 17 



61- » 
28*53 



59 4< 
30*19 



59 la &8*42 

30*49 Sl'lft 



58 12 
31*48 



5741 
32*19 



57'8 56*3^5^1 
32*52 33*24 33*59 



SSJ26 54*50 
34*34 35*10 



C0i5 
a945 



5948 69 ai 

30*12 30' 



39 31 



5852 
8 



58*9^ 57 54 
3I*2« 32*6 



57*36 57*6 
32*» 32*54 



572» 
32*37 



56 
33^8 



5i|56i9 
33*41 



34*.s 



34*49 



54 si 53*S6 
35*2S 36*2 
53-45 53*8 



59 29 

30*31 



59*2 



58'yi58 



30 se 3126 31' 



56*2 
33* 



55*30 54*56 54*2.' 



2* 3358 34 30 35 4^ 



35 39 36 IB 



3652 37 



;jli 389 
St^ 51*0 
38*20 39 C 
SO'si SoCit 
39*9 39*4 
50*4.' 49^ 
39*56 40*31 

^[74^ 



58 44 58 



i 16 57^46 
1*44 32*12 



57*19 
32*41 



56 49 56 19 



33*11 



3341 



5547 

34*13 



55*1! 



S4*4i' 



54*7 
3S*S3 



5332 
36*28 



52*S2 52*18 
37*6 37*42 



58' 
32*0 



57*32 57*3' 
32*28 32*57 



5633 

33*27 



56 3 55 
33*S7' 34*28 



350 



54» 
35*3i 



S3*S4| 53*261 52''44J 52* 8 
36*6 



36 40 



37 16 



5l''3<i 
37^62 38*30 

&l°20 5<f43 

38*40 39* .7 
50*35 48" 



5716 
32*^ 



5648 56*19 
\i. 33*41 



5549 
34*11 



55*18 54*47 
34*4a3s' 



54i5 

35*45 



S3 42 

36*18 



53*8 
36*S2 



52-33 
37*27 



3t S6"4: 



33 28 33 



- . SS*L 

3*56 34'a 



^21 54"'k 
34*39 35' 



34*58 



5434 SS* 
35*26 35" 



53*S^5i 
36' 



53*30 
sel 36*30 



5223 

37*37 



51-47 
38*13 



•56 49' 



3848 3915 40 A 



43 41^ 
48M47 
4l*2t 42 
47*48 47 



5551 
34*9 



9^*39 i^ 
35*21 35* 



5423 

3^37' 



3»42 



52^ 
37*.4 



5i"38 sr 



3 

38*57 



5027 
39*33 



4949 

40 



49 II 
40*40 



559 
34*51 



I5s8^ 
' 2 36'*- 



52; 
SJ 37* 



24 37 



52 3 

57 



5l2»| 

38*3 



50 54 
39*6 



so 19 
39* 



49 
40« 



S 

40-65 



48*17 
41*33 



4a S3 



54*29 53" 



tBS258 



°2b Sr 

"34 38* 



46°2i 47*44 47° & 



46*8* 



38 40 39 14 3948 40 24 4 



4^^143 
46'2i 
43°33 | 44 
4^*6 . 
44*ao^^ 



41 



5348 S3 17 



52*48 



36)2 



3643 37 12 



5216 
37*44 



1*45 SI 
>*15 



5I''4 5^ 
^•6 39' 



3848 39 



50 39 

21 



50 s 
39*55 



30 48 54 48 



17 

I4r43 



474« 
142*2^ 



ATT 
42*59 



42 



53*8' 
36*52 



•a 37* 



51*36 



32 49 



52 3824 38 



47*16 



4659 



46*26 



40 34,41114147 4214 43 1 43 40 



26 



BROWN & SHARPE MFG. CO. 



TABLE 3. 
Angle of Face. 



41 



40393837363534333231 



^ 



30232827 



12 



I3S7| 
70 



13 57 

70 3 i 



14 IB 

70*6' 



14 39' 
69" 



15 £4: 

68*32 



15*9- 
L67V9 



16 ti 
67\i 



17*3 

65i3 



1827 

6 3' 3' 



20 * 
62' 9' 



13 



15*17 



\5'i» 
68* 



I6'85 
67*43 



6'5I 

67*9 



17 

6633 



T74T 

65*56 



18*- 
64s4j 



63*5 



20 
62*14 



21 
6l°a3 



21 54 
602i 



14 



16 "34 

6a°o 



16 59 

6 729 



17 
66V6 



17 50 
66ge 



IB 
65Wt\ 



18 45 

65*8 



19 
64*30 



!9* 



20 to 
63*6 



206 



69' 



23.J8 

5850 



15 



T7\. 
fc7'« 



I 7*55 

664-5 



18 4-4 

6540 



I9"i 
65^3 



19 
64''g* 



20 
634* 



2044J 

6 5°a 



21 
6 2*4 



2 



23 lo 
60°2 



2 5*S( 
59 *a 



t4-3S\ 

58m 



25 to 
57J4. 



16 



I8°4e 
66°. 



19 

6533 



I9°3J 
64^ 



63° 



2E°9 

6145 



60V 



58<>a 



24*8 
s|58s4 



25 

57io 



26"',i 
56^2 



27' 
55*8 



17 



I9V« 
64' 



2024 
64'eo 



20 
634d 



21 °e, 



Zl s» 
62 3 



22 <4 
61 so 



2257 

6l''9 



23j3 

6015 



24 

59Vo 



26 I* 
57 



27i9 
56'; 3 



2 7^7 



18 



63*9 



22°6 
62*S4 



6(54 



23°« 

61*17 



2344] 
60V 



24 ,e 



24«« 

59% 



£54 
58°io 



2615 

573. 



26 57 
56*39 



SS%t\ 



274i 

5544 



2819 
54 



29 I. 
53 50 



*0s 

52*47 



31*40 
5(i4 



19 



22" 
623« 



22 4! 

62*1 



eo2 



23 

61*44 



3°32 
60V* 



60* 



Ii5 

58*54J 



25* 



2537 

58*37 



2 6°. 5 
57*5 



£6'54 
57*4 



27°38 

56' 



29"s 
54.***l 



2 9 56 

53^ 



3O4.3 
52*68 






20 



Z3 

6130 



24 I 

o 
60 53 



243* 25*6 



26*16 
5 6*10 



2645 
5 725 



27»4 
56 



28(5 
55^9 



28"58 
5458 



30 3. 
53*9 



33 '» 
50*4 



24*39 

60lts 



as 

5946 



26°53 
574 



2 7 '30 
57*C 



2850 
S5\* 



2931 

54*36 



30 
53*8 



31°* 
52*40 



3(°5a 

51*52 



32*^ 
50°J5 



33*36 



34a. 



22 



59W 



26°s 
5B°4 



2653 

5a 



27 27 
57*9 



28°s 
56*3*1 



28*3 

55*5 



29 21 



30 5 
54*7 



30 
53 26 



313* 

S2°a» 



34"3 
494. 



34 57 
48 3 7 



35*54 

47I2 



23 



26 52 

58*16 



2 7 2 

5736 



28*0 

56*56 



28>6 
56*(4 



29 

5530 



2953 

54*43 



35*35 

53*57 



53*6 



32*1 
Sz'is 



32 

5/ '2*1 



33"34 
50*28 



34°27 

49*i« 



35 20 



36.5 

4-72tU46'2o 
171^38*2 



24 



2757 

57t5 



2944 
55 



30''Z2 
54*26 



3r2 
53^ 



31"*. 
52*5 



32*281 

52*2 



33 "14 
^(*o 



34 
49tol 



36°3i| 

1471, 



25 



£8 
56*14 



29 34 

55 



SO'ii 

54*52 



30 
54"» 



*3 3 



29 
53*23 



32 

5 2 '36 



32°52 

51 



33 

50*57 



50*5 



36°6 

48*14 



37*7| 

A6*.5 



38°4 4 
4 5 



26 



30^ 
54*34JS352 



31 54 

53°8 



3?q 

52 



33 
6/35 



33'i>8 
50 



344$ 

49 



35 3. 
49*3 



36.9 
4 8 



37.0 
47*2 



3a"2 
46°. i 



38 46 

45*0 



39 54 
44*r 



4052 

4.3*2 



27 



54*19 



31 39 
53°37 



32 16 
5 2°«4 



32 57 

5 2% 



33 3 

5I°2: 



34 sc 

5034 



3ST7 
49%7| 



35"3 
49*5 



354S 
48*5$ 



36 3t 
4^\ 



3725 
A 7*7 



39 .0 

4>S 10 



41 I 

4 a"* 



28 



32*2 

53*22 



32°39 

52*S9 



33°.8 

5IS6 



33 57 
5 



3439 



36"7 
48*7 



36 "5 2 

4756 



375^ 
46 



29 



30 



31 



32 



33 



35 



36", 



37 



38 



3259 
52*27 



33 3«| 
51 



3458 

50*6 



35°39 
49*29 



56^ 
t48*4 



37 8 
4750 



39%^ 40^4 47V 
45*.o44*2 43j* 



4^ 



33°57 
51*34 



3436 

50*50 



35 15 
50*7 



3556 

49*2 



36"3e 3 

48 34 47i4K6"'55l46''3 



3941 
45*9 



403214124 
45' 



3453 

50°*, 



35*6 
4950 



353 

49*47| 



36' 

49", 3| 



'52 37 



35 
47S9 



3820 
4632 



39 5 
46° I 



3952 
4-6 10 



40°. 



4I°»2 



42-« 



37"6 
48*2 z 



3748 
•473 4 



383 

46°4d 



39"i5 
4 5'* 9 



40° I 

4S°» 



4049 
44' 



4138 
43*24 



364 9 
46*49 



38 

4646 



39 26 



wl 4.8*,, 



38"ii 

47*2 7 



3&°5J 

46*4S 



39°3S 

45*57 



40°l8 
45*8 



4r4 
442«l43\9 



o 
4724 



39°3 395M40°t6 
46's9l45*54 45*6 



4241 



39S2 
45*52 



40^4115 



4 2*0 

43*34 



4a- 



40'0 

4552 



40*0 
45*8 



4ii2 

44*22 



42°5 
43*37 



40"47j 
45*7 



4ll 

44-°24| 



42"9 
43*s*t 



39: 



40 

41 «■ 



4a.'.e 

43*2 



42 58 



PROVIDENCE, R. I. 



27 



TABLE 3. — {Continued.^ 



Angle of Face. 



26|25|242322 



2019 



18 



16 



IS 



14 



13 



12 



•2 
13 
11 
15 






2 13. 



Z4'3 
5 8' 2 56V 9 



25*s Zb" 



25 2 
55°3Z 



26*3 
54*7 



Z7°u 

52 39 



28 IS 
5/* 3 



23Vi 



3 
47is 



32; 
4-5°24] 



34 26 
43' 



3*3(6 

40*iC 



38'n 



22 37 
59°2 9 



23'26 

58 ;b 



24*^5 
5 7*2, 



2 7°6 
533 6 






30 *t 



333* 



35"ic 
4-3°2i 



3o"55 
4-I°9 



38<< 



24 IS 
574.9 



23 lb 

5646 



26°8 

55°38J 



28 4. 
53°6 



50 



32^2 
47 8 



34 e 
45°2 



3550 
4-32 



3728 



26 
56° 3 



27 3 
S5°y 



iTsa 
55°S6 



Z&'SB 

52 



32*,9 

48' 



333 < 
4 7*c 



34ff« 
45' 



Je-ia 



37i7 



17 
]8 

13 
20 

21 
22 
23 
24 
25 
26 



2 94.31 

S2°2 



30V» 
S 



3l°«o 

49 46 



32 "58 

4S°2J 



34-(2 
46*S2 



3531 
h4«S°i9 



36'3^38°2l 
43' 



2 9 "so 



30a6| 
52 "o 



3126 

50«s 



32''28 
49°32 



36°o 
45°s 



3i"5 

5141 



32^2 

50°32 



33% 

4 9° 8 



34*8 

48°2 



4641 



36'28 
45*6 



374* 39' 
43°4j42 



32°36| 
50° 8 



3336 
49°B 



34 3B 
K7«4 



3549I3653 

46 "36 



39°24 
42''fo 



35°6 
4-8 J7K746 



36°8 

46°32J 



3&2* 

43°S2 



35 3 
4739 



36 '32 
46°28 



3737 

15° 3 



3844 39''54 
43°S6 42''34 



3 9 "as 

4-E'*27 



3652 

46°24. 



375 51 
45' 



39"o 40 a 
4358 4240 



38.2 
45° t 



A020 

4246 



39 29 40 32 



4l3B 



40' 
42^7 



4-146 



41 53 



g, = 90° - {a, + y5) 
g, = 90° - («, + /3) 

(See page ij.) 



28 



BROWN & SHARPS MFG. CO. 



NATURAL SINE. 



Deg. 


0' 


10' 


20' 


33' 


40' 


50' 


CO' 







.00000 


.00291 


.00581 


.00872 


.01163 


.01454 


.01745 


89 


1 


.01745 


.02030 


.02326 


.02617 


.02908 


.03199 


.03489 


i 88 


3 


.03489 


.03780 


.04071 


.04361 


.04652 


.04943 


.05233 


87 


3 


.05283 


.05524 


.05814 


.06104 


.06395 


.06685 


.06975 


86 


4 


.06975 


.07265 


.07555 


.07845 


.08135 


.08425 


.08715 


85 


5 


.08715 


.09005 


.09295 


.09584 


.09874 


.10163 


.10452 


84 


6 


.10452 


.10742 


.11031 


.11320 


.11609 


.11898 


.12186 


83 


7 


.1218G 


.12475 


.12764 


.13052 


.13341 


.13629 


.13917 


82 


8 


13917 


.14205 


.14498 


.14780 


.15068 


.15356 


.15643 


1 81 


9 


.15643 


15930 


.16217 


.16504 


.16791 


.17078 


.17364 


80 


10 


.17364 


.17651 


.17937 


.18223 


.18509 


.18795 


.19080 


79 


11 


.19080 


.19366 


.19651 


.19936 


.20221 


.20506 


.20791 


78 


12 


.20791 


.21075 


.21359 


.21644 


.21927 


.22211 


.22495 


77 


13 


.22495 


.22778 


.23061 


.23344 


.28627 


.23909 


.24192 


76 


14 


.24192 


.24474 


.24756 


.25088 


.25319 


.25600 


.25881 


75 


15 


.25881 


.26162 


.26443 


.23723 


.27004 


.27284 


.27563 


74 


16 


.27568 


.27843 


.28122 


.28401 


.28680 


.28958 


.29237 


73 


17 


.29287 


.29515 


.29798 


.30070 


.30347 


.30624 


.30901 


72 


18 


.30901 


.31178 


.81454 


.31733 


.32006 


.32281 


.82556 


71 


19 


.32556 


.82881 


.33106 


.83380 


.33654 


.38928 


. 34202 


70 


20 


.34202 


.84475 


.34748 


.35020 


.35298 


.85565 


.35836 


69 


21 


.35836 


.36108 


.86879 


.86650 


.86920 


.37190 


.37460 


68 


22 


.37460 


.377S0 


.87999 


.88268 


.38536 


.38805 


.39078 


67 


23 


.39073 


.39340 


.89607 


.39874 


.40141 


.40407 


.40673 


66 


24 


.40673 


.40989 


.41204 


.41469 


.41733 


.41998 


.42261 


65 


25 


.42261 


.42525 


.42788 


.43051 


.43313 


.43575 


.43887 


64 


26 


.48887 


.44098 


.44859 


.44619 


.44879 


.45139 


.45899 


63 


27 


.45399 


.45658 


.45916 


.46174 


.46432 


.46690 


.46947 


62 


28 


.46947 


.47203 


.47460 


.47715 


.47971 


.48226 


.48481 


61 


29 


.48481 


.48785 


.48989 


.49242 


.49495 


. 49747 


.50000 


i 60 


30 


.50000 


.50251 


.50503 


.50753 


.51004 


.51254 


.51503 


i 59 


31 


.51508 


.51752 


.52001 


.52249 


.52497 


.52745 


.52991 


! 58 


32 


.52991 


.53238 


.53484 


.53780 


.53975 


.54219 


.54463 


! 57 


33 


.54468 


.54707 


.54950 


. 55193 


.55486 


.55677 


.55919 


1 56 


34 


.55919 


.56160 


.56400 


.56640 


. 56880 


.57119 


.57357 


! 55 


35 


.57357 


.57595 


.57833 


.58070 


.58306 


.58542 


.58778 


54 


36 


.58778 


.59013 


.59248 


.59482 


.59715 


.59948 


.60181 


53 


37 


.60181 


.60413 


.60645 


.60876 


.61106 


.61336 


.61566 


52 


38 


.61566 


.61795 


.62023 


.62251 


.62478 


.62705 


.62932 


1 51 


39 


.62932 


.63157 


.63383 


.63607 


.63832 


.64055 


.64278 


1 50 


40 


.64278 


.64501 


.64723 


.64944 


.65165 


.65386 


.65605 


' 49 


41 


.65605 


.65825 


.66043 


.66262 


.66479 


.66696 


.66913 


48 


42 


.66913 


.67128 


.67344 


.67559 


.67773 


.67986 


.68199 


47 


43 


.68199 


.68412 


.68624 


. 68835 


.69046 


.69256 


.69465 


46 


44 


.69465 


.69674 


.69883 


.70090 


.70298 


.70504 


.70710 


45 




60' 


50' 


40' 


30' 


20' 


10' 


0' 


Deg. 



NATURAL COSINE. 



PROVIDENCE, R. 



29 



NATURAL SINE. 



Deg. 


1 

0' 


10' 


20' 


30' 


40' 


SO' 


60' 




45 


.70710 


.70916 


.71120 


.71325 


.71528 


.71731 


.71934 


44 


46 


.71934 


.72185 


.72336 


.72537 


.72737 


.72930 


.73135 


43 


47 


.73185 


. 73333 


. 73530 


.73727 


.73923 


.74119 


.74314 


42 


48 


.74314 


.74508 


.74702 


.74895 


.75088 


.75279 


.75471 


41 


49 


.75471 


. 75661 


.75851 


.76040 


.76229 


.76417 


.76604 


40 


50 


.76604 


.76791 


. 769T7 


.77162 


.77347 


.77531 


.77714 


89 


m 


.77714 


.77897 


.78070 


.78260 


.78441 


.78621 


.78801 


38 


63 


.78801 


.78979 


.79157 


.79335 


.79512 


.79688 


.79863 


1 37 


53 


.79863 


.80038 


.80212 


.80385 


.80558 


.80730 


.80901 


36 


54 


.80901 


.81072 


.81242 


.81411 


.81580 


.81748 


.81915 


35 


55 


.81915 


.82081 


.82247 


.82412 


.82577 


.82740 


.82903 


34 


56 


.82903 


.83066 


.83227 


.83388 


.83548 


.83708 


.83867 


33 


57 


.83867 


.84025 


.84182 


.84389 


.84495 


.84650 


.84804 


1 32 


58 


.84804 


.84958 


.85111 


.85264 


.85415 


.85566 


.85716 


31 


59 


.85716 


.85866 


.86014 


.86162 


.88310 


.86456 


.86602 


30 


60 


.86602 


.86747 


.86892 


.87035 


.87178 


.87320 


.87462 


29 


61 


.87462 


.87602 


.87742 


.87881 


.88020 


.88157 


.88294 1 


28 


62 


.88294 


.88430 


.88566 


.88701 


.88835 


.88968 


.89100 ! 


27 


63 


.89100 


.89232 


.89363 


.89493 


.89622 


.89751 


.89879 


26 


64 


.89879 


.90006 


.90132 


.90258 


.90383 


.90507 


.90630 


25 


65 


.90630 


.90753 


.90875 


.90996 


.91116 


.91235 


.91354 


24 


66 


.91354 


.91472 


.91589 


.91706 


.91821 


.91936 


.92050 


23 


67 


.92050 


.92163 


.9227G 


.92388 


.92498 


.92609 


.92718 


22 


68 


.92718 


. 92827 


.92934 


.93041 


.93148 


.93253 


.93358 


21 


69 


.93358 


.93461 


.93565 


.93667 


.93768 


.93869 


.93909 


20 


70 


.93969 


.94068 


. 94166 


.94264 


.94360 


.94456 


.94551 


19 


71 


.94551 


.94640 


.94739 


.94832 


.94924 


.95015 


.95105 1 


18 


73 


.95105 


.95195 


.95283 


.95371 


.95458 


.95545 


.95630 t 


17 


73 


.95630 


.95715 


.95799 


.95882 


.95964 


.96045 


.96126 


16 


74 


.96126 


.96205 


.96284 


.96363 


.96440 


.96516 


.96592 


15 


75 


.96592 


.96667 


.96741 


.96814 


.96887 


.96958 


.97029 


14 


76 


.97029 


.97099 


.97168 


.97237 


.97304 


.97371 


.97437 


13 


77 


.97437 


.97502 


.97566 


.9762:) 


.97692 


.97753 


.97814 


12 


78 


97814 


.97874 


.97934 


.97992 


.98050 


.98106 


.98162 : 


11 


79 


.98162 


.98217 


.98272 


.98325 


.98378 


.98429 


.98480 


10 


80 


.98480 


.98530 


.98580 


.98628 


.98670 


.98722 


.98768 


9 


81 


.98768 


.98813 


.98858 


.98901 


.98944 


.98985 


.99026 


8 


82 


.99026 


.99066 


.09106 


.99144 


.99182 


.99218 


.99254 ! 


7 


83 


.99254 


.99289 


.99323 


.99357 


.99389 


.99421 


.99452 


6 


84 


.99452 


.99482 


.99511 


.99539 


.99567 


.99593 


.99619 


5 


85 


.99619 


. 99644 


.99668 


.99691 


.99714 


.99735 


.99756 


4 


86 


.99756 


.99776 


.99795 


99813 


.99830 


.99847 


.99863 


3 


87 


.99863 


.99877 


.99891 


.99904 


.99917 


.99928 


.99939 


2 


88 


.99939 


.99948 


.99957 


.99965 


.99972 


.99979 


.99984 


1 


89 


.99984 


.99989 


.99993 


.99996 


.99998 


.99999 


1.0000 1 







60' 


50' 


40' 


30' 


20' 


10' 


0- ! 


Deg. 



NATURAL COSINE. 



3° 



BROWN & SHARPE MFG. CO. 



NATUEAL TANGENT. 



Ueg. 


0' 


10' 


£0' 


30' 


40' 


SO' 


60' 







.00000 


.00290 


.00581 


.00872 


.01163 


.01454 


.01745 


89 


1 


.01745 


.02036 


.02327 


.02618 


.02909 


.03200 


.03492 


88 


3 


.03492 


.03783 


.04074 


.04366 


.04657 


.04949 


.05240 


87 


3 


.05240 


.05532 


.05824 


.06116 


.06408 


.06700 


.06992 


86 


4 


. .06992 


.07285 


.07577 


.07870 


.08162 


.08455 


.08748 


85 


5 


.08748 


.09042 


.09335 


.09628 


.09922 


.10216 


.10510 


84 


6 


.10510 


.10804 


.11099 


.11393 


.11688 


.11983 


.12278 


83 


7 


.12278 


.12573 


.12869 


.13165 


.13461 


. 13757 


.14054 


82 


8 


.14054 


.14350 


.14647 


.14945 


.15242 


.15540 


.15838 


81 


9 


.15838 


.16136 


. 16435 


.16734 


.17033 


.17332 


.17632 


80 


10 


.17632 


. 17932 


.18233 


.18533 


.18834 


.19136 


.19438 


79 


11 


.19438 


.19740 


.20042 


.20345 


.20648 


.20951 


.21255 


78 


12 


.21255 


.21559 


.21864 


.22169 


.22474 


.22780 


.23086 


77 


13 


.23086 


.23393 


.23700 


.24007 


.24315 


.24624 


.24932 


76 


14 


.24932 


.25242 


.25551 


.25861 


.26172 


.26483 


.26794 


75 < 


15 


.26794 


.27106 


.27419 


.27732 


.28046 


.28360 


.28674 


74 


16 


.28674 


.28989 


.29305 


.29621 


.29938 


.30255 


.30573 


73 


17 


.30573 


.30891 


.31210 


.31529 


.31850 


.32170 


. 32492 


72 


18 


.32492 


.82813 


.33136 


.33459 


.33783 


.34107 


.34432 


71 


19 


.34432 


.34753 


.35084 


.35411 


.35739 


.36067 


.36397 


70 


20 


.36397 


.36726 


.37057 


.37388 


.37720 


.38053 


.38386 


69 


21 


.38386 


.38720 


.39055 


.39391 


.39727 


.40064 


.40402 


68 


22 


.40402 


.40741 


.41080 


.41421 


.41762 


.42104 


.42447 


67 


83 


.42447 


.42791 


.43135 


.43481 


.43827 


.44174 


.44522 


66 


24 


.44522 


.44871 


.45221 


.45572 


.45924 


.46277 


.46630 


65 


25 


.46630 


.46985 


.47341 


.47697 


.48055 


.48413 


.48773 


64 


26 


.48773 


.49133 


.49495 


.49858 


.50221 


.50586 


.50952 


63 


27 


.50952 


.51319 


.51687 


.52056 


.52427 


.52798 


.53170 


62 


28 i 


.53170 


.53544 


.53919 


.54295 


.54672 


.55051 


.55430 


61 


29 ; 


.55430 


.55811 


.56193 


.56577 


.56961 


.57347 


.57735 


60 


30 1 


.57735 


.58123 


.58513 


.58904 


.59297 


.59690 


.60086 


59 


31 


.60086 


.60482 


.60880 


.61280 


.61680 


.62083 


.62486 


58 


32 


.62480 


.62892 


.63298 


.63707 


.64116 


.64528 


.64940 


57 


33 


.64940 


.65355 


.65771 


.66188 


.66607 


.67028 


.67450 


56 


34 


.67450 


.67874 


.68300 


.68728 


.69157 


.69588 


.70020 


55 


35 


.70020 


.70455 


.70891 


.71329 


.71769 


.72210 


.72654 


54 


36 


.72654 


.73099 


.73546 


.73996 


.74447 


.74900 


.75355 


53 


37 


.75355 


.75812 


.76271 


.76732 


.77195 


.77661 


.78128 


52 


38 


.78128 


.78598 


.79069 


.79543 


.80019 


.80497 


80978 


51 


39 


.80978 


.81461 


.81946 


.82433 


.82923 


.83415 


.83910 


50 


40 


.83910 


.84406 


.84900 


.85408 


.85912 


.86419 


.86928 


49 


41 


.86928 


.87440 


.87955 


.88472 


.88992 


.89515 


.90040 


48 


42 


.90040 


.90568 


.91099 


.91633 


.92169 


.92709 


.93251 


47 


43 


.93251 


.93796 


.94345 


.94896 


.95450 


.96008 


.96568 


46 


44 


.96568 


.97132 


.97699 


.98269 


.98843 


.99419 


1.0000 


45 


- 


60' 


50 


40' 


30 


20' 


10' 


0' 


Deg. 



NATURAL COTANGENT. 



PROVIDENCE, R. I. 



31 



NATURAL TANGENT. 



Deg. 


0' 


10' 


20' 


30' 


40' 


50' 


60 




45 


1.0000 


1.0058 


1.0117 


1.0176 


1.0235 


1.0295 


1.0355 


44 


46 


1.0355 


1.0415 


1.0476 


1.0537 


1.0599 


1.0661 


1.0723 


43 , 


47 


1.0723 


1.0786 


1.0849 


1.0913 


1.0977 


1 . 1041 


1.1106 


42 


48 


1.1106 


1.1171 


1.1236 


1.1302 


1.1369 


1.1436 


1.1503 


41 


49 


1.1503 


1.1571 


1.1639 


1.1708 


1.1777 


1.1847 


1.1917 


40 


50 


1.1917 


1.1988 


1.2059 


1.2131 


1.2203 


1.2275 


1.2349 


39 


51 


1.2349 


1.2422 


1.2496 


1.2571 


1.2647 


1 . 2723 


1.2799 


38 


53 


1.2799 


1.2876 


1.2954 


1.3032 


1.3111 


1.3190 


1.3270 


37 


53 


1.3270 


1.3351 


1.3432 


1.3514 


1.3596 


1.3680 


1.3763 


36 


54 


1.3763 


1.3848 


1.3933 


1.4019 


1.4106 


1.4193 


1.4281 


35 


55 


1.4281 


1.4370 


1.4459 


1.4550 


1.4641 


1.4733 


1.4825 


34 


56 


1.4825 


1.4919 


1.5013 


1.5108 


1.5204 


1.5301 


1.5398 


33 


57 


1.5398 


1.5497 


1.5596 


1.5696 


1.5798 


1.5900 


1.6003 


32 


58 


1.6003 


1.6107 


1.6212 


1.6318 


1.6425 


1.6533 


1.6642 


31 


59 


1.6642 


1.6753 


1.6864 


1.6976 


1 . 7090 


1.7204 


1.7320 


30 


60 


1.7320 


1.7437 


1.7555 


1.7674 


1.7795 


1.7917 


1.8040 


29 


61 


1.8040 


1.8164 


1.8290 


1.8417 


1.8546 


1.8676 


1.8807 


28 


62 


1.8807 


1.8940 


1.9074 


1.9209 


1.9347 


1.9485 


1.9626 


27 


63 


1.9626 


1.9768 


1.9911 


2.0056 


2.0203 


2.0352 


2.0503 


26 


64 


2.0503 


2.0655 


2.0809 


2.0965 


2.1123 


2.1283 


2.1445 


25 


65 


2.1445 


2.1609 


2.1774 


2.1943 


2.2113 


2.2285 


2.2460 


24 


66 


2.2460 


2.2637 


2.2816 


2.2998 


2.3182 


2.3369 


2.3558 


23 


67 


2.3558 


2.3750 


2.3944 


2.4142 


2.4342 


2.4545 


2.4750 


22 


68 


2.4750 


2.4959 


2.5171 


2.5386 


2.5604 


2.5826 


2.6050 


21 


69 


2.6050 


2.6279 


2.6510 


2.6746 


2.6985 


2.7228 


2.7474 


20 


70 


2.7474 


2.7725 


2.7980 


2.8239 


2.8502 


2.8770 


2.9042 


19 


71 


2.9042 


2.9318 


2.9600 


2.9886 


3.0178 


3.0474 


3.0776 


18 


72 


3.0776 


3.1084 


3.1397 


3.1715 


3.2040 


3.2371 


3.2708 


17 


73 


3.2708 


3.3052 


3.3402 


3.3759 


3.4123 


3.4495 


3.4874 


16 


74 


3.4874 


3.5260 


3.5655 


3.6058 


3.6470 


3.6890 


3.7320 


15 


75 


3.7320 


3.7759 


3.8208 


3.8667 


3.9136 


3.9616 


4.0107 


14 


76 


4.0107 


4.0610 


4.1125 


4.1653 


4.2193 


4.2747 


4.3314 


13 


77 


4.3814 


4.3896 


4.4494 


4.5107 


4.5736 


4.6382 


4.7046 


12 


78 


4.7046 


4.7728 


4.8430 


4.9151 


4.9894 


5.0658 


5.1445 


11 


79 


5.1445 


5.2256 


5.3092 


5.3955 


5.4845 


5.5763 


5.6712 


10 


80 


5.6712 


5.7693 


5.8708 


5.9757 


6.0844 


6.1970 


6.3137 


9 


81 


6.3137 


6.4348 


6.5605 


6.6911 


6.8269 


6.9682 


7.1153 


8 


82 


7.1153 


7.2687 


7.4287 


7.5957 


7.7703 


7.9530 


8.1443 


7 


83 


8.1443 


8.3449 


8.5555 


8.7768 


9.0098 


9.2553 


9.5143 


6 


84 


9.5143 


9.7881 


10.078 


10.385 


10.711 


11.059 


11.430 


5 


85 


11.430 


11.826 


12.250 


12.706 


13.196 


13.726 


14.300 


4 


86 


14.300 


14.924 


15.604 


16.349 


17.169 


18.075 


19.081 


3 


87 


19.081 


20.205 


21.470 


22.904 


24.541 


26.431 


28.636 


2 


88 


28.636 


31.241 


34.367 


38.188 


42.964 


49.103 


57.290 


1 


89 


57.290 


68.750 


85.939 


114.58 


171.88 


343.77 


00 







60 


50 


40' 


30' 


20' 


10' 


0' 


Deg. 



NATURAL COTANGENT. 



32 



BROWN & SHARPE MFG. CO. 



chapte:r IV. 
WORM AND WORM WHEEL. 

(Fig. 8.) 




33 



FORMULAS. 

L = lead of worm. 

N = number of teeth in gear. 

m = threads per inch in worm. 

d= diameter of worm. 
d' = diameter of hob. 
T = throat diameter. 
B = blank diameter (to sharp corners). 
C = distance between centers. 

= thickness of hob-slotting^ cutter. 

/= width of bands at bottom. 

If = pitch circumference of worm. 

V = width of worm thread tool at end. 
w = width of worm thread at tap. 
P = diametral pitch. 
P' = circular pitch. 

s = addendum. 

t = thickness of tooth at pitch line. 
/" = normal thickness of tooth. 
/= clearance at bottom of tooth. 
D" = working depth of tooth. 
D" + /= whole depth of tooth. 

d = angle of thread with axis. 
If the lead is for single, double, triple, etc., thread, then 
L = P', 2 P; 3 P', etc. 



34 BROWN & SHARPE MFG. CO. 



cx 


= 60 to 90' 


L 


_ I 




m 


P' 


ttT 




N + 2 


D 


_ NP' _ 




n 


T 


-V" 



b= n {d — 2 s) 

o. _ L j Practical only when width of wheel on wheel pitch circle 
~^ \ is not more than % pitch diameter of worm. 



^n, 


= /COS (J 




r^ ■ 


2 




r' 


= r' + D-+/ 




C 


2 




B 

= 


= T4-2(r^-r^ 

. -335 P' + .^^ 
2 


-f) ^ 


/ = 


^D'' + 2/+r 




d' 


= ^+ 2/ 




V - 


= =31 P' 
- .335 P' 





measurement of sketch is generally 
sufficient. 



Note.— The notations and formulas referring to tooth parts, given on page 5 for 
spur gears, apply to worm wheels, and are here used. 

Note. — Hob and worm should be marked, as per example : 
4 threads per V single .25 P'; .25 L. 
2 threads per i" double .25 P'; .50 L. 



PROVIDENCE, R. I. 



35 



UNDERCUT IN WORM WHEELS. 



In worm wheels of less than 30 teeth the thread of the worm 
(being 29°) interferes with the flank of the gear tooth. Such 
a wheel finished with a hob will have its teeth undercut. To 
avoid this interference two methods may be employed. 

First Method. — y[3k& throat diameter of wheel 



T = cos 



14/2 






or 



T = -937 N ^ 



4-f 



This formula increases the throat diameter, and conse- 
quently the center distance. The amount of the increase can 
be found by comparing this value of T with the one as obtained 
by formula on page 34. To keep the original center distance, 
the outside diameter of the worm must be reduced by the 
same amount the throat diameter is increased. 

Second Method. — Without changing any of the dimensions 
we found by the formulas given on page 34, we can avoid the 
interference to be found in worm wheels of less than 30 teeth 
by simply increasing the angle of worm thread. We find the 
value of this angle by the following formula : 
Let there be 

2 y = angle of worm thread. 

N = number of teeth in worm wheel. 





cos y - 


= / 


-^ 












rom this formula we obtain the following values : 


N 


29 


28 


27 


26 


25 


24 


23 


22 


21 


20 


2 y 


30K 


31 


31^ 


32X132^ 


1 
33/2 34% 


35 


36 


37 


N 


19 


18 


17 


16 


15 


14 


13 


12 




2 y 


38 


39 


40 


41^ 


42^ 


44 >^ 


46M 


48 



As this latter formula involves the making of new hobs in 
many cases, on account of change of angle, we prefer to reduce 
the diameter of worm as indicated by first method, if the dis- 
tance of centers must be absolute. 



36 



BROWN & SHARPE MFG. CO. 



CHARTER V. 



SPIRAL OR SCREW GEARING. 

(Figs. 9, lo, II.) 




Fig. 9, 



In spiral gearing the wheels have cylindrical pitch surfaces, 
but the teeth are not parallel to the axis. The line in which 
the pitch surface intersects the face of a tooth is part of a 
screw line, or helix, drawn at the pitch surface. A screw 
wheel may have one or any number of teeth. A one-toothed 
wheel corresponds to a one-threaded screw, a many-toothed 
wheel to a many- threaded screw. The axes may be placed at 
any angle. 

Consider spiral gears with : 

I. Axes parallel. 
II. Axes at right angles. 
III. Axes any angle. 



PROVIDENCE, R. I, 



37 




Fig, 10, 

Let there be : 

xt" ~ ^ number of teeth in gears ] ^ 

C = center distance. 

P = diametral pitch 

P' = circular pitch. 
P" = normal diametral pitch. 
P'" = normal circular pitch. 

y = angle of axes. 

Lj = exact lead of spiral on pitch surface. 
L^ = approximate lead of spiral on pitch surface. 

T = number of teeth marked on cutter to be used when 
teeth are to be cut on milling machine. 

D = pitch diameter. 

B = blank diameter. 

a = ) 
" _ I angle of teeth with axis 

/= thickness of tooth. 
s = addendum. 
D" + /= whole depth of tooth. 

Note. — Letters a and d occurring at bottom of notations refer to gears a and f>. 



I. — Axes Parallel. 
Gears of this class are called twisted gears. The angle of 
teeth with axes in both gears must be equal and the spirals 
run in opposite directions. The angles are generally chosen 
small (seldom over 20°) to avoid excessive end thrust. End 
thrust may, however, be entirely avoided by combining two 
pairs of wheels with right and left-hand obliquity. Gears of 
this class are known as Herringbone gears. They are com- 
paratively noiseless running at high speed. 



SS BROWN & SHARPE MFG. CO. 

II. — Axes at Right Angles. 
Here we must always have : 

1. The teeth of same hand spiral ; 

2. The normal pitches equal in both gears ; and 

3. The sum of the angles of teeth with axes = 90°. 

Choosing Angle of Teeth with Axes. 

1. If in a pair of gears the ratio of the number of teeth is 
equal to the direct ratio of the diameters, /. e., if the number of 
teeth in the two gears are to each other as their pitch diame- 
ters, then the angles of the spirals will be 45° and 45° ; for, this 
condition being fulfilled, the circular pitches of the two gears 
must be alike, which is only possible with angles of 45°. In 
such a combination either gear may be the driver. 

2. If the ratio of the diameters determined upon is larger 
or smaller than the ratio of the number of teeth, then the 
angles are : 

In such gears the velocity ratio is measured by the number 
of teeth, and not by the diameters. 

3. Given N^^, N,, and C : 

If P„' is made = P^,', then we have case '' i " and 

But if Pa is assumed, then : 

p,_ C7r->^ N,P,^ 

and 

tan a^ = —^ tan aj, = -A 

a p , o p r 

The gear whose P' or a is larger will be the driver, on 
account of the greater obliquity of the teeth. 
4. Given N„, N^, and C or D. 
See case " 7 " under III., considering ;/ = 90°. 

III. — Axis at any Angle (y). 

5. Given case " i," under II., then angles of spirals = }4 yt 
for the same reason. 

6. Analogous cases to "2" and "3," under II., may be 
worked out, when angles of axes = y, but they have been 



PROVIDENCE, R. 



39 



omitted, partly because the formulas are too cumbersome, and 
partly because they are to some extent covered by cases " 5 " 

and "7." 

7. Given N„, N,, and C, or one of the pitch diameters. We 
find the angles by a graphic method, which for all practical 
purposes is accurate enough ; ro and v are the axes of gears 
forming angle ;/ (see diagram. Fig. 11.) On these axes v^e 
lay off lines r and o v representing the ratio of the number 
of teeth (velocity ratio), so that N« : N^, : : r j : j z;, and 




FUj. 11, 



construct parallelogram r s v. Then, according to Mc- 
Cord,* the angles formed by the tangent i- ^ in the pitch con- 
tact with the axes of the gears insures the least amount of 
sliding. In bisecting angle y by tangent u o and using angles 
produced in this manner we equally distribute the end thrust on 
both shafts. Both methods have their advantages ; to profit 
by both we select angles a^ and o',,, produced by tangent x, 
bisecting angle u o s. 

Thus we have when angles are found and C given, 

p,,i _ 2 C TT cos OL^ cos O'ft 

" N^ cos a^^ -f- Nft cos a^ 
and when D^ given 

pm ^ Dg n cos a^ ^^^ 



D.= 



It cos OTj, 



* McCord, Kinematics, page 378. 



40 brown & sharpe mfg. co. 

General Formulas. 
y = a^ + a^ 

Pfn TD tn 
a = ^h 

P' N P'" N 
D = or = 



7t 7t cos a 

B = D + 2j or = D + ^ 

pn 

P' =r or 



N cos a 

P« = JL (Pitch of cutter.) 

p'n ^ ^ 



p/n J 






2 






lO 

cos ^a 


(See Note i.) 




J N P' 

Lj = or = 

tan a 


N7t 

or = 

Ptan<af 


N P- 
tan fl' cos a 


y lO W G, 

SG, 


{See Note 2.) 




/cos 45 
ycos' 45 


°= .707iT\ 

° = .50 ; 





No. 5 cutter for T from 


21 to 25 


U 5 ii U «4 <l 


17 to 20 


C< 7 .. U u .< 


14 to 16 


U 8 <t (( (( l( 


12 to 13 



Note i. — Cutters of regular involute system 
Use No. 1 cutter for T from 135 up. 

2 " " " " 55 to 134 

u 2 .. c, u .c 2- to ^^ 

" 4 " " " " 26 to 34 

Note 2. — Gears used on spiral head and bed for Brown & Sharpe milling 
machine : 

W = number of teeth in gear on worm. 
G. = " " ist " stud. 

G2 = " " 2d " stud. 

S = " " " screw. 

Should a spiral head of different construction be used, the formula would not 
apply. 



PROVIDENCE, R. I. 41 



CHAPTER VI. 

INTERNAL GEARING. 

PART A.— INTERNAL SPUR GEARING. 
(Figs. 12, 13, 14, 15, 16.) 

A little consideration will show that a tooth of an internal 
or annular gear is the same as the space of a spur — external 
gear. 

We prefer the epicycioidal form of tooth in this class of 
gearing to the involute form, for the reason that the difficulties 
in overcoming the interference of gear teeth in the involute 
system are considerable. Special constructions are required 
when the difference between the number of teeth in gear and 
pinion is small. 

In using the system of epicycioidal form of tooth in which 
the gear of 15 teeth has radial flanks, this difference must be 
at least 15 teeth, if the teeth have both faces and flanks. Gears 
fulfilling this condition present no difficulties. Their pitch 
diameters are found as in regular spur gears, and the inside 
diameter is equal to the pitch diameter, less twice the adden- 
dum. 

If, however, this difference is less than 15, say 6, or 2, or i, 
then we may construct the tooth outline (based on the epicy- 
cioidal system) in two different ways. 

First Method. — To explain this method better, let us sup- 
pose the case as in Fig. t2, in which the difference between 
gear and pinion is more than 15 teeth. Here the point o of 
the describing circle B (the diameter of which in the best 
practice of the present day is equal to the pitch radius of a 15 
tooth gear, of the same pitch as the gears in question) gene- 
rates the cycloid o, o\ o*, o^, etc., when rolling on pitch circle 
L L of gear, forming the face of tooth ; and when rolling on 
the outside of L L the flank of the tooth. In like manner is the 
face and flank of the pinion tooth produced by B rolling out- 
side and inside of E E (pitch circle of pinion). A little study 



42 



BROWN & SHARPE MFG. CO. 



of Fig. 12 (in which the face and flank of a gear tooth are 
produced) will show the describing circle B divided into 12 




equal parts and circles laid through these points (i, 2, 3, etc.), 
concentric with L L. We now lay off on L L the distances 
o-T, 1-2, 2-3, etc., of the circumference of B, and obtain points 



PROVIDENCK, R. I. 



43 



i', 2\ 3\ etc. [Ordinarily it is sufficient to use the chord.] It 
will now readily be seen that B in rolling on L L will success- 
ively come in contact with i', 2\ 3^ etc., c meanwhile moving 
to <:\ tr', c^^ etc. (points on radii through i', 2', 3', etc.), and the 
generating point o advancing to o^, o^ o^, etc., being the inter- 
sections of B with c\ r^, r', etc., as centers and the circles laid 
through I, 2, 3, etc. Points o, o\ o^ o^, etc., connected with a 
curve give the face of the tooth ; in like manner the flank is 
obtained. 

In this manner the form of tooth is obtained, when the 
difference of teeth in gear and pinion is less than 15, with the 
exception that the diameter of describing circle B 

^ - n\ 



-y^-^) 



where P = diametral pitch, N and n number of teeth in gears. 
The distances of the tooth above and below the pitch line 
as well as the thickness t are determined as in regular spur 
gears by the pitch, except when the difference in gear and 
pinion is very small, where we obtain a short tooth, as in Figs. 
13 and 14. In such a case the height of tooth is arbitrary and 
only conditioned by the curve. In internal gears it is best to 
allow more clearance at bottom of tooth than in ordinary spur 
gears. 



29 Teeth 




42 T. 



8 P. 



30 Teeth 



FifJ. 13. 




In a construction of this kind it is suggested to draw the 
tooth outline many times full size and reduce by photography. 
An equally multiplied line A B will help in reducing. 



44 



BROWN & SHARPE MFG. CO. 




PROVIDENCE, R. I. 45 

Second Method. — The difference between gear and pinion 
being very small, it is sometimes desirable to obtain a smooth 
action by avoiding what is termed the " friction of approach- 
ing action."* This is done, the pinion drivings by giving gear 
only flanks, Fig. 15, and the gear drivings by giving gear only 
faces, Fig. 16. In both these cases we have but one describ- 
ing circle, whose diameter is equal to the difference of the two 
pitch diameters. The construction of the curve is precisely 
the same as described under A. The describing circle has 
been divided into 24 parts simply for the sake of greater 
accuracy. 



PART B.— INTERNAL BEVEL GEARS. 

(Fig. 17.) 

The pitch surfaces of bevel gears are cones whose apexes 
are at a common point, rolling upon each other. The tooth 
forms for any given pair of bevel gears are the same as for a 
pair of spur gears (of same pitch) whose pitch radii are equal to 
the respective apex distances of the normal cones (/. ^., cones 
whose elements are perpendicular upon the elements of the 
bevel gear pitch cones). (Compare Fig 19, page 50.) 

The same is true of internal bevel gears, with the modifica- 
tion that here one of the pitch cones rolls inside of the other. 
The spur gears to whose tooth forms the forms of the bevel 
gear teeth correspond, resolve themselves into internal spur 
gears (Fig. 17). The problem is now to be solved as indicated 
in the first part of this chapter. 



* McCord, Kinematics, pages icy, 108. 



46 



BROWN & SHARPE MFG. CO. 



S r. 

Gear 40 Treth 
Finion 20 Teeth 




Fig. 17. 



PROVIDENCE, R. I. 47 



CHAPTER VII. 

GEAR PATTERNS. 

(Fig. 18.) 

To place in bevel gears the best iron where it belongs, the 
tooth side of the pattern should always be in the nowel, no 
matter of what shape the hubs are. 

Hubs, if short, may be left solid on web ; if long they should 
be made loose. A long hub should go on a tapering arbor, to 
prevent tipping in the sand. i° taper for draft on hubs when 
loose, and 3° when solid is considered sufficient. 

Coreprints as a rule are made separate, partly to allow the 
pattern to be turned on an arbor, partly for convenience, 
should it be desirable to use different sizes. 

Put rap- and draw-holes as near to center as possible. 
Referring to Fig. 18, make L = D for D from ^" to i}4", or 
even more, should hubs be very long. Otherwise if D is more 
than i)^" leave L = i^". 

Iron pattern before using should be marked, rusted and 
vi^axed. 

Shrinkage — For cast-iron, Y^" per foot. 
For brass, yV " 

Cast-iron gears^ especially arm gears, do not shrink Yz" per 
foot. In making iron patterns the following suggestions have 
been found useful : 

Up to 12" diameter allow no shrink. 
From 12" to 18" " " ^ regular shrink. 

" 18" to 24" " '' >4 " 

" 24" to 48" " " Yi 

Above 48" " '' .10" " 

for cast-iron. 



48 



BROWN & SHARPE MFG. CO. 




PROVIDENCE, R. I. 



49 



If in gears the teeth are to be cast, the tooth thickness t in 
the pattern is made smaller than called for by the pitch, to avoid 
binding of the teeth w^hen cast. No definite rule can be given, 
as the practice varies on this point. For the different diam- 
etral pitches w^e vi^ould advise making / smaller by an amount 
expressed in inches, as given in the following table : 



DiAM 


Pitch. 


Amount t 
IS Smaller. 


DiAM. Pitch. 


Amount t 
IS Smaller. 




i6 


.oio' 


5 


.020" 




12 


OI2" 


4 


.022" 




lO 


.014" 


3 


.026" 




8 


.016'' 


2 


• 030" 




6 


.018" 


I 


.040" 



50 



BROWN & SHARPE MFG. CO. 



CHAPTER VIII. 

DIMENSIONS AND FORM FOR BEVEL GEAR 

CUTTERS. 

(Fig. 19.) 

The data needed to determine the form and thickness of a 
bevel gear cutter are the following : 
P = pitch. 

N = number of teeth in large gear. 
n= number of teeth in small gear. 
F = length of face of tooth, measured on pitch line. 
After having laid out a diagram of the pitch cones a b c and 
a b f^ and laid off the width of face, the problem resolves itself 
into two parts : 

Part I. — Determine Proper Curve for Cutter. 
It willbe remembered that in the involute system of cutters 
(the only one used for bevel gears that are cut with rotary 
cutter), a set of eight different cutters is made for each 
pitch, numbering from No, i to No. 8, and cutting from 
a rack to 12 teeth. Each number represents the form of 
a cutter suitable to cut the indicated number of teeth. For 
instance. No. 4 cutter (No. 4 curve) will cut 26 to 34 teeth. 
In order to find the curve to be used for gear and pinion 
we simply construct the normal pitch cones by erecting 
the perpendicular p q through /;, Fig. 19. We now measure the 
lines b q and b p^ and taking them as radii, multiplying each by 
2 and P we obtain a number of teeth for which cutters of 
proper curves may be selected. From example we have : 

Gea7' : b q — g^" ; 2 X P X 9.75 = 97 T No. 2 curve. 

Pinion: b p = 3;^" ; 2 X P X 3.5 = 35 T No. 3 curve. 
The eight cutters which are made in the involute system 
for each pitch are as follows : 

No. I will cut wheels from 135 teeth to a rack. 



2 




a i 


' 55 


(( 




134 teeth. 


3 




>< < 


' 35 


li 




54 " 


4 




(( (, 


' 26 


<( 




34 '' 


5 




it ( 


' 21 


(( 




25 " 


6 




u ( 


^7 


(( 




20 " 


7 




a i 


' 14 


<< 




16 " 


8 




U ( 


* 12 


« 




13 " 



PROVIDENCE, R. I. 



51 




52 BROWN & SHARPE MFG. CO. 

Part II. — Determine Thickness of Cutter. 

It is very evident that a bevel gear cutter cannot be thicker 
than the width of the space at small end of tooth ; the practice 
is to make cutter .005" thinner. Theoretically the cutting angle 
{h) is equal to pitch angle less angle of bottom {or h = a — fi'). 
Practically, however, better results are obtained by making 
h = a — fi (substituting angle of top for angle of bottom), and 
in calculating the depth at small end, to add the full clearance 
(/) to the obtained working depth, giving equal amount of 
clearance at large and small end. This is done to obtain a 
tooth thinner at the top and more curved. As the small end 
of tooth determines the thickness of cutter, we shall have to 
find the tooth part values at small end. From the diagram it 
will be seen that the values at large end are to those at small 
end as their respective apex distances {a b and a I). The 
numerical values of these can be taken from the diagram and 
the quotient of the larger in the smaller is the constant where- 
with to multiply the tooth values at large end, to obtain those 
at small end. In our example we find : 



« / = 3.8 


.655 = 


= constant 


For 5 P we have 


^=.3141 






/' = .2o57 




s = .2000 






/ = .i3io 




/=.o3i4 






/=o3i4 




^ +/=.23i4 




s 


+ /=.i624 




•"+/=. 4314. 




D" 


/ = .T3IO 

+ /=-2934 





From the foregoing it is evident that a spur gear cutter 
could not be used, since a bevel gear cutter must be thinner. 

If in gears of more than 30 teeth the faces are proportion- 
ately long, we select a cutter w^hose curve corresponds to the 
midway section of the tooth. The curve of the cutter is found 
by the method explained in Part I. of this Chapter. 



PROVIDENCE, R. L 53 



Chai^ter IX. 

DIRECTIONS FOR CUTTING BEVEL GEARS 
WITH ROTARY CUTTER. 

(Fig. 20.) 

In order to obtain good results, the gear blanks must be of 
the right size and form. The following sizes for each end of 
the tooth must be given the workman : 
Total depth of tooth. 
Thickness of tooth at pitch line. 
Height of tooth above pitch line. 
These sizes are obtained as explained in Chapter VIII. 
The workman must further know the cutting angle (see 
(formula on page 13 and compare Chapter VIII.), and be pro- 
vided with the proper tools with which to measure teeth, etc. 
In cutting a gear on a universal milling machine the opera- 
tions and adjustments of the machine are as follows : 

1. Set spiral bed to zero line. 

2. Set cutter central with spiral head spindle. 

3. Set spiral head to the proper cutting angle. 

4. Set the index on head for the number of teeth to be cut, 
leaving the sector on the straight or numbered row of holes, 
and set the pointer (or in some machines the dial) on cross-feed 
screw of millmg machine to zero line. 

5. As a matter of precaution, mark the depth to be cut for 
large and small end of tooth on their respective places. 

6. Cut two or three teeth in blank to conform with these 
marks in depth. The teeth will now be too thick on both their 
pitch circles. 

7. Set the cutter off the center by moving the saddle to or 
from the frame of the machine by means of the cross-feed 
screw, measuring the advance on dial of same. The saddle 
must not be moved further than what to good judgment 



54 



BROWN & SHARPE MFG. CO. 




IH.-. 



Fig. 20, 



PROVIDENCE, R. I. 55 

appears as not excessive ; at the same time bearing in mind 
that an equal amount of stock is to be taken off each side of 
tooth. 

8. Rotate the gear in the opposite direction from which the 
saddle is moved off the center, and trim the sides of teeth (A) 
(Fig. 20.) 

9. Then move the saddle the same distance on the opposite 
side of center and rotate the gear an equal amount in the 
opposite direction and trim the other sides of teeth (C). 

10. If the teeth are still too thick at large end E, move the 
saddle further off the center and repeat the operation, bearing 
in mind that the gear must be rotated and the saddle moved 
an equal amount each way from their respective zero settings. 

It is generally necessary to file the sides of teeth above the 
pitch line more or less on the small ends of teeth, as indicated 
by dotted lines F F. This applies to pinions of less than 30 
teeth. 

For gears of coarser pitch than 5 diametral it is best to 
make one cut around before attempting to obtain the tooth 
thickness. 

The formulas for obtaining the dimensions and angles of 
gear blanks are given in Chapter III. 



56 BROWN & SHARPE MFG. CO. 



CHAPTER X. 

THE INDEXING OF ANY WHOLE OR FRAC- 
TIONAL NUMBER. 

(Fig. 21.) 




•CJinnge Gear 

Fig. 21. 



In indexing on a machine the question simply is : How 

many divisions of the machine index have to be advanced to 

advance a unit division of the number required. To which 

is the 

divisions of machine index 

answer = 

number to be indexed 

Suppose the number of divisions in index wheel of machine 
to be 216. 



Example I. — Index 72. 
Answer: 216 



3 (3 turns of worm), 

72 



PROVIDENCE, R. I. 57 

Example II. — Index 123. 

— = I + -93 
123 123 

If now we should put on worm shaft a change gear having 
123 teeth, give the worm shaft. Fig. 21, one turn, and in addi- 
tion thereto advance 93 teeth of the change gear (to give the 
fractional turn), we would have indexed correctly one unit of 
the given number, and so solved the problem. Should we not 
have change gear 123 we may try those on hand. The ques- 
tion then is : How many teeth (j) of the gear on hand (for 
instance 82) must we advance to obtain a result equal to the 
one when advancing 93 teeth of the 123 tooth gear? We have : 

^ = — where ;^ = 62 
123 82 

Example III. — Index 365, change gear 147. 

— = -^ where ;^ = 87 — -3_ 
365 147 365 

Here 147 is the change gear on hand. In indexing for a unit 
of 365 we advance 8 teeth of our 147 tooth gear. It is evident 
that in so doing we advance too fast and will have indexed 
three teeth of our change gear too many when the circle is 
completed. To avoid having this error show in its total amount 
between the last and the first division, we can distribute the 
error by dropping one tooth at a time at three even intervals. 



Example IV. 


— Index ] 


[90. 
















216 
190 


= I ■ 


^.6 

190 




Change gear 


on 


hand 


90 


T 




26 
190 


90 


where 


X 


190 











To distribute the error in this case we advance one addi- 
tional tooth at a time of the change gear at six even intervals. 

Example V. — Index 117.3913. 

216 _ 986087 



117.3913 1173913 

This example is in nowise different from the preceding 
ones, except that the fraction is expressed in large numbers. 
This fraction we can reduce to lower approximate values, 
which for practical purposes are accurate enough. This is 
done by the method of continued fractions. [For an explana- 



58 BROWN & SHARPE MFG. CO. 

tion of this method we refer to our " Practical Treatise on 
Gearing."] 

986087 





II739I3 


986087) 


"^^'' 




187826) 986087 (5 
939130 




46957) 187826 (3 
140871 




46955) 46957 Ci 
46955 




2) 46955 (23477 
46954 




I) 2 (2 
2 



986087 _ ^ 
II739I3 ^ - 



+ 1 



5+ 1 

3 + 1 

i + i 



23477 + 1 
2 



^=3 I 23477 



ar=i ^ = 5 d =. \6 21 493033 986087 
«' = i <^' = 6 ^' = 19 25 586944 1173913 

Note.— Find the first two fractions by reduction = - and — j — = ^ ; the 

II I + I 6 



5 



others are then found by the rule \ , ^ ' ^ 



The fraction \\ is a good approximation; putting therefore 
a change gear of 25 teeth on worm shaft, we advance (beside 
the one full turn) 21 teeth to index our unit. 

Of course, in using any but the correct fraction we have an 
error every time we index a division ; so that when indexed 
around the whole circle, we have multiplied this error by the 
number of divisions. 

In the present example this error is evidently equal to the 
difference between the correct and the approximate fraction 
used. Reducing both common fractions to decimal fractions 
we have : 

-^ — = .84000006 

1173913 

21 o 

— = .84000000 . 1_ J« • • 

21- —^ = error in each division. 

■^ .00000006 



PROVIDENCE, R. I. 59 

.00000006 ' 1 17.3913 = .00000703348 total error in complete 
circle. This error is expressed in parts of a unit division. (To 
find this error expressed in inches, multiply it by the distance 
between two divisions, measured on the circle.) In this case 
the approximate fraction being smaller than the correct one, 
in indexing the whole circle we fall short .00000703348 of a 
division. 



Example VI.- 


— Index 15.708 




216 _ J, _^ 11796 




15.708 " 15708 




ii796_ 983 




15708 1309 




983)1309(1 




983 




326) 983 (3 
978 

5) 326 (65 






30 




26 




25 




1)5(5 




5 









983 I 








^309 , + i 




3+1 




65 + 1 




5 




I 3 65 5 




I 3 196 983 




I 4 261 1309 



In using the approximation |^f J the error for each division 
(found as above) will be .000002917, for the whole circle 
.0000458. In this case, the approximation being larger than 
the correct fraction, we overreach the circle by the error. 



6o 



BROWN & SHARPE MFG. CO. 



CHAPTKR XI. 

THE GEARING OF LATHES FOR SCREW 
CUTTING. 

(Figs. 22, 23.) 

The problem of cutting a screw on a lathe resolves itself into 
connecting the lathe spindle with the lead screw by a train of 
gears in such a manner that the carriage (which is actuated by 




Simple Gearing, 

Fig. 22. 



PROVIDENCE, R. I. 



6j 



the lead screw) advances just one inch, or some definite dis- 
tance, while the lathe spindle makes a number of revolutions 
equal to the number of threads to be cut per inch. 

The lead screw has, w4th the exception of a very few cases, 
always a single thread, and to advance the carriage one inch it 
therefore makes a number of revolutions equal to its number 




Compound Gearing- 

Fig. 23. 



of threads per inch. Should the lead screw have double 
thread, it will, to accomplish the same result, make a number 
of revolutions equal to half its number of threads per inch. It 
follows that we must know in the first place the number of 
threads per inch on lead screw. 



62 BROWN & SHARPE MFG. CO. 

It ought to be clearly understood that one or more inter- 
mediate gears, which simply transmit the motion received from 
one gear to another, in no wise alter the ultimate ratio of a 
train of gearing. An even number of intermediate gears 
simply change the direction of rotation, an odd number do not 
alter it. 

The gearing of a lathe to solve a problem in screw cutting 
can be accomplished by 

A. Simple gearing. 

B. Compound gearing. 

Referring to the diagrams, Figs. 22 and 23, we have in Fig. 
22 a case of simple, and in Fig. 23 a case of compound gear- 
ing. 

In simple gearing the motion from gear E is transmitted 
either directly to gear Ron lead screw or through the interme- 
diate F. In compound gearing the motion of E is transmitted 
through two gears (G and H) keyed together, revolving on the 
same stud n, by which we can change the velocity ratio of the 
motion while transmitting it from E to R. With these four 
variables E, G, H, R, we are enabled to have a wider range of 
changes than in simple gearing. 

B and C, being intermediate gears, are not to be considered. 
If, as is generally the case, gear A equals gear D, we disregard 
them both, simply remembering that gear E (being fast on 
same shaft with E) makes as many revolutions as the spindle. 
Sometimes gear D is twice as large as gear A, then, still con- 
sidering gear E as making as many revolutions as the spindle, 
we deal with the lead screw as having twice as many threads 
per inch as it measures. 



SIMPLE GEARING. 

Let there be : the number of teeth in the different gears 
expressed by their respective letters, as per Fig. 22, and 

s — threads per inch to be cut, 
L ■— threads per inch on lead screw ; then 

I. "^ =: 5: 

L D 



PROVIDENCE, R. I. 6^ 

If now one of the two gears D and R is selected, the other 
will be : 

R=iP; D = t^ 

L s 

2. The two gears may be found by making 

p. ~"^ T [-where/ may be any number. 

3. The above holds good when a fractional thread is to be 
cut, but if the fraction is expressed in large numbers, as, for 
instance, s = 2.833 (2t¥oV)' we first reduce this fraction (y^o¥o) to 
lower approximate values by the process of continued fraction 
(see pages 57 and 58). 



) ICOO (1 

833 

167) 813 (4 

668 




165) 167 (i 
165 
2) 165 (82 
16 

~ 5 
4 

1)2(2 
2 





I 4 I 82 


2 


145 414 

I 5 -6 497 


833 

1000 


^ = .S^^ (nearly) and s : 



= 25 
6 


£ L = 4, and we select D 


= 48 


R = i^ R=34 





COMPOUND GEARING. 

4. In a lathe geared compound for cutting a screw the 
product of the drivers (E and H, Fig. 23) multiplied by the num- 
ber of threads to be cut must equal the product of the driven 
(G and R) multiplied by the number of threads on lead screw. 
This is expressed by 

E.H.j=G.R.Lor ^—^-^ = i 
G. R .L 



64 BROWN & SHARPE MFG. CO. 

If three of the gears E, H, G, R have been selected, the 
fourth one would be either 



E — 


GR L 




H s 


H- 


G R L 




E s 


G- 


.E Hi- 




R L 


R = 


E H^ 
GL 




RGL 



or 



or 



or 



/^^G_X 

Vl.e.h/ 



E H 

If a fractional thread is to be cut, as under "3," we reduce 
the fraction to lower approximate values. 

Example. — Gear for 5.2327 threads per inch, lead screw is 
6 threads. 





.2327 


_ 2327 

lOOOO 




2327) lOOCO (4 

9308 
692) 2327 (3 

2076 

251) 692 (2 
502 

190) 251 (1 
190 
61) 19c (3 

183 

7) 61 (8 






5)7(1 

5 

2; 5 (2 

4 

1)2(2 

2 









4 


3213 


8122 


I 
4 


3 7 10 37 
13 30 43 159 


306 343 992 2327 
1315 1474 4263 lOOOO 




— = .2327 (nearly) and 5.2327 = s— 
43 43 



Selecting E = 43, H = 52, R = 50, and 

^ E.H.J , ^ 43 . ';2 . t;i4 

G = we have G = ^ — 5 i£l = ^9. 

R . L 50 . 6 '^^ 



PROVIDENCE, R. I. 65 

5. The examples so far given all deal with single thread. 
The pitch of a screw is the distance from center of one thread to 
the center of the next. The lead of a screw is the advance for 
each complete revolution. In a single thread screw the pitch 
is equal to the lead, while in a double thread screw the pitch 
is equal to one-half the lead ; in a triple thread screw equal to 
one-third the lead, etc. 

If we have to gear a lathe for a many-threaded screw 
(double, triple, quadruple, etc.), we simply ascertain the lead, 
and deal with the lead as we would with the pitch in a single 
thread screw, /. ^., we divide one inch by it, to obtain the num- 
ber of threads for which we have to gear our lathe. 

Example. — Gear for double thread screw, lead = .4654. 
Number of threads per inch to be geared for is : 

— L_=r _!_= 2.1487 
Lead -4654 

Lead screw is four threads per inch. 
As in previous examples, we reduce the fraction .i4S'j=-^-^-^-^ 
to lower approximate values by the process of continued frac- 
tion. 

From the different values received in the usual way we 
select : 

ij = .1487 (nearly) and 2.1487 = 2|^ 

We have therefore : 

Selecting -^ G = 30 
(h=4o 

G . L 30 . 4 

Note. — In using any but the original fraction we commit an error. This error 
can be found by reducing the approximate fraction used to a decimal fraction, and 
comparing it with the original fraction. In the above example the original fraction is 

.1487 and 
H = . 14864 



Error = .00006 inch in lead. 

In cutting a multiple screw, after having cut one 
thread, the question arises how to move the thread tool the 
correct amount for cutting the next thread. 



66 BROWN & SHARPE MFG. CO. 

In cutting double, triple, etc., threads, if in simple or com- 
pound gearing the number of teeth in gear E is divisible by 
2, 3, etc., we so divide the teeth ; then leaving the carriage 
at rest we bring gear E out of mesh and move it forward one 
division, whereby the spindle will assume the correct position. 

Is E not divisible we find how many teeth (V) of gear R 
are advanced to each full turn of the spindle. Dividing this 
number by 2 for double, by 3 for triple thread, etc., we 
advance R so many teeth, being careful to leave the spindle at 
rest. 

We have for simple gearing : 

R 

for compound gearing : 

V- EH 
G.R 

If in simple gearing both E and R are not divisible, one 
remedy would be to gear the lathe compound ; or the face- 
plate may be accurately divided in two, three or more slots, 
and all that is then necessary is to move the dog from one slot 
to another, the carriage remaining stationary. 



m'sy 



::^ 



